{"x402Version":2,"name":"ScorpTech x402 API","description":"ScorpTech x402 API: pay-per-call tools for agents: web, security, data","network":"solana:5eykt4UsFv8P8NJdTREpY1vzqKqZKvdp","asset":"USDC","payTo":"Gzn6fbrn11cd61iH4mgQvtmf4PHmeMCRn8msU4hjpty8","scheme":"exact","openapi":"https://scorptech.it.com/api/x402/openapi.json","llms":"https://scorptech.it.com/api/x402/llms.txt","mcp":"https://scorptech.it.com/api/x402/mcp","operator":{"name":"ScorpTech","url":"https://scorptech.it.com"},"resources":[{"method":"GET","path":"/v1/headers-audit","url":"https://scorptech.it.com/api/x402/v1/headers-audit","summary":"Graded HTTP security-header audit of a URL: HSTS, CSP quality, XFO, nosniff, Referrer/Permissions/COOP/CORP/COEP, cookie","price":"$0.005","params":["url"]},{"method":"GET","path":"/v1/email-auth","url":"https://scorptech.it.com/api/x402/v1/email-auth","summary":"Email authentication & deliverability posture for a domain: SPF (qualifier, lookup count, duplicates), DMARC policy/rua/","price":"$0.005","params":["domain"]},{"method":"GET","path":"/v1/domain-intel","url":"https://scorptech.it.com/api/x402/v1/domain-intel","summary":"One-call domain due-diligence: RDAP registration age/expiry/registrar, A/AAAA/NS/MX/CAA, live TLS certificate (issuer, e","price":"$0.01","params":["domain"]},{"method":"GET","path":"/v1/robots-sitemap","url":"https://scorptech.it.com/api/x402/v1/robots-sitemap","summary":"Fetch and parse robots.txt plus discovered sitemaps for a domain: user-agents, allow/disallow samples, sitemap index vs ","price":"$0.005","params":["domain"]},{"method":"GET","path":"/v1/tech-stack","url":"https://scorptech.it.com/api/x402/v1/tech-stack","summary":"Lightweight technology fingerprint from public HTML/headers: CMS, frameworks (Next/Nuxt/React/Vue), CDNs, analytics, pay","price":"$0.005","params":["url"]},{"method":"GET","path":"/v1/rdap-age","url":"https://scorptech.it.com/api/x402/v1/rdap-age","summary":"Lightweight RDAP registration age lookup: registered/expiry dates, ageDays, daysToExpiry, registrar, status flags via rd","price":"$0.005","params":["domain"]},{"method":"GET","path":"/v1/phishing-score","url":"https://scorptech.it.com/api/x402/v1/phishing-score","summary":"Heuristic phishing-likelihood score for a URL/domain: brand typosquat distance, subdomain depth, punycode, RDAP age, HTT","price":"$0.008","params":["url"]},{"method":"GET","path":"/v1/cve-lookup","url":"https://scorptech.it.com/api/x402/v1/cve-lookup","summary":"Search NIST NVD CVE API 2.0 by product/keyword with optional CVSSv3 severity filter. Returns CVE ids, scores, vectors, d","price":"$0.01","params":["product","severity"]},{"method":"GET","path":"/v1/package-risk","url":"https://scorptech.it.com/api/x402/v1/package-risk","summary":"npm or PyPI package risk/typosquat check: registry metadata, weekly downloads (npm), deprecation, license/repo presence,","price":"$0.01","params":["ecosystem","name"]},{"method":"GET","path":"/v1/github-health","url":"https://scorptech.it.com/api/x402/v1/github-health","summary":"GitHub repository health summary via public API: stars/forks/issues, license, languages, last push age, latest release &","price":"$0.008","params":["repo"]},{"method":"GET","path":"/v1/opengraph","url":"https://scorptech.it.com/api/x402/v1/opengraph","summary":"Extract Open Graph, Twitter Card, title/description, canonical, icons and JSON-LD @types from a public URL. Ideal for li","price":"$0.005","params":["url"]},{"method":"GET","path":"/v1/url-markdown","url":"https://scorptech.it.com/api/x402/v1/url-markdown","summary":"Fetch a public HTML page and convert main/article content to Markdown (Turndown). Scripts/styles stripped; output capped","price":"$0.005","params":["url"]},{"method":"POST","path":"/v1/schema-validate","url":"https://scorptech.it.com/api/x402/v1/schema-validate","summary":"Validate JSON data against a JSON Schema (Ajv + formats). Body: { \"schema\": {...}, \"data\": {...} }. 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Detect ","price":"$0.005","params":["url"]},{"method":"GET","path":"/v1/http-probe","url":"https://scorptech.it.com/api/x402/v1/http-probe","summary":"Fast HTTP GET probe: final URL, status, timing, response bytes, and a subset of interesting headers (server, cache, HSTS","price":"$0.005","params":["url"]},{"method":"GET","path":"/v1/x402-probe","url":"https://scorptech.it.com/api/x402/v1/x402-probe","summary":"x402 preflight for AI agents: inspect any paid endpoint BEFORE paying. 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Options: ecc L|M|Q|H, size (px/m","price":"$0.005","params":["text","ecc","size","margin","format","fg","bg"]},{"method":"GET","path":"/v1/text-stats","url":"https://scorptech.it.com/api/x402/v1/text-stats","summary":"Text statistics: characters, words, unique words, sentences, paragraphs, reading/speaking time, Flesch readability (Engl","price":"$0.005","params":["text","wpm","top"]},{"method":"POST","path":"/v1/html-to-markdown","url":"https://scorptech.it.com/api/x402/v1/html-to-markdown","summary":"Convert an HTML string to clean Markdown (GFM tables, fenced code, nested lists). No fetch. Body: { \"html\": \"...\", \"base","price":"$0.005","params":[]},{"method":"POST","path":"/v1/image-info","url":"https://scorptech.it.com/api/x402/v1/image-info","summary":"Inspect an image without decoding pixels: format, dimensions (EXIF-orientation aware), megapixels, animation, sha256, EX","price":"$0.005","params":[]},{"method":"POST","path":"/v1/image-resize","url":"https://scorptech.it.com/api/x402/v1/image-resize","summary":"Resize / convert an image (GD): width/height up to 4096, fit contain|cover|fill|inside, output png|jpeg|webp|gif, qualit","price":"$0.01","params":[]},{"method":"GET","path":"/v1/sec-edgar-search","url":"https://scorptech.it.com/api/x402/v1/sec-edgar-search","summary":"SEC EDGAR filings search: latest 10-K/10-Q/8-K (or any form) for a company by name, ticker or CIK, with direct document ","price":"$0.01","params":["company","q","forms","from","to","limit"]},{"method":"GET","path":"/v1/sec-form4","url":"https://scorptech.it.com/api/x402/v1/sec-form4","summary":"Insider trading from SEC Form 4: the latest Form 4 filings for a ticker or CIK parsed from EDGAR ownershipDocument XML. ","price":"$0.01","params":["company","limit","type"]},{"method":"GET","path":"/v1/company-profile","url":"https://scorptech.it.com/api/x402/v1/company-profile","summary":"Company profile from its own website: name, description, logo, social profiles, contact emails/phones, postal address, s","price":"$0.03","params":["url"]},{"method":"GET","path":"/v1/weather-nws","url":"https://scorptech.it.com/api/x402/v1/weather-nws","summary":"US weather from the National Weather Service for a latitude/longitude: 7-day (or hourly) forecast periods with temperatu","price":"$0.01","params":["lat","lon","units","hourly"]},{"method":"GET","path":"/v1/arxiv-search","url":"https://scorptech.it.com/api/x402/v1/arxiv-search","summary":"Search arXiv papers: title, authors, full abstract, categories, dates, DOI/journal ref and direct abs + PDF links. Plain","price":"$0.01","params":["query","category","sort","limit","start"]},{"method":"GET","path":"/v1/earthquakes","url":"https://scorptech.it.com/api/x402/v1/earthquakes","summary":"Recent earthquakes from the USGS: filter by bounding box or center point + radius, minimum magnitude and time window (la","price":"$0.01","params":["minlat","maxlat","minlon","maxlon","lat","lon","radiusKm","minmag","days","start","end","limit","order"]},{"method":"GET","path":"/v1/holidays-business-days","url":"https://scorptech.it.com/api/x402/v1/holidays-business-days","summary":"Public holidays for 100+ countries (optionally one state/region) and business-day math in one call: list holidays for a ","price":"$0.01","params":["country","op","year","date","days","start","end","subdivision","weekend"]},{"method":"GET","path":"/v1/hn-search","url":"https://scorptech.it.com/api/x402/v1/hn-search","summary":"Search Hacker News stories and comments (Algolia HN index): by relevance or newest first, with date range, story/comment","price":"$0.01","params":["query","type","sort","from","to","author","minPoints","limit","page","exact"]},{"method":"GET","path":"/v1/dns-propagation","url":"https://scorptech.it.com/api/x402/v1/dns-propagation","summary":"DNS propagation check: resolves one record type for a domain across Cloudflare 1.1.1.1, Google 8.8.8.8 and Quad9 9.9.9.9","price":"$0.01","params":["domain","type","resolvers"]},{"method":"GET","path":"/v1/ssl-cert-info","url":"https://scorptech.it.com/api/x402/v1/ssl-cert-info","summary":"Inspect a live TLS certificate: subject, issuer, SANs, validity window, days to expiry, SHA-256 fingerprint, key type/si","price":"$0.01","params":["host","port"]},{"method":"GET","path":"/v1/email-validate","url":"https://scorptech.it.com/api/x402/v1/email-validate","summary":"Validate an email address without sending mail: RFC 5321/5322 syntax, MX/A lookup (null-MX aware), disposable-domain det","price":"$0.005","params":["email","check_mx"]},{"method":"POST","path":"/v1/password-strength","url":"https://scorptech.it.com/api/x402/v1/password-strength","summary":"Offline password strength audit (POST only, so the password never appears in a URL or access log; never echoed or stored","price":"$0.005","params":[]},{"method":"POST","path":"/v1/validate-batch","url":"https://scorptech.it.com/api/x402/v1/validate-batch","summary":"Offline format validator for up to 100 values: email, URL, IPv4/IPv6, domain, UUID (version/variant), credit card (Luhn ","price":"$0.005","params":[]},{"method":"GET","path":"/v1/sol-balance","url":"https://scorptech.it.com/api/x402/v1/sol-balance","summary":"SOL and SPL token balances (classic SPL Token + Token-2022) for a Solana wallet address, via public Solana RPC. Read-onl","price":"$0.005","params":["address","include_zero","include_token2022","limit"]},{"method":"GET","path":"/v1/sol-token-meta","url":"https://scorptech.it.com/api/x402/v1/sol-token-meta","summary":"Solana SPL / Token-2022 mint metadata: decimals, total supply, mint & freeze authority, Token-2022 extensions, and name/","price":"$0.005","params":["mint","include_market"]},{"method":"GET","path":"/v1/sol-token-risk","url":"https://scorptech.it.com/api/x402/v1/sol-token-risk","summary":"Heuristic risk screen for a Solana token mint: mint/freeze authority, Token-2022 extensions (permanent delegate, transfe","price":"$0.01","params":["mint","include_holders"]},{"method":"GET","path":"/v1/evm-balance","url":"https://scorptech.it.com/api/x402/v1/evm-balance","summary":"Native ETH and ERC-20 token balances for an address on Ethereum, Base or Arbitrum via public RPC. Defaults to major toke","price":"$0.005","params":["address","chain","tokens","include_defaults","include_zero"]},{"method":"GET","path":"/v1/evm-tx-info","url":"https://scorptech.it.com/api/x402/v1/evm-tx-info","summary":"Decode an EVM transaction (Ethereum, Base, Arbitrum): status, block, timestamp, confirmations, from/to, value, gas used,","price":"$0.005","params":["hash","chain"]},{"method":"GET","path":"/v1/fx-rate","url":"https://scorptech.it.com/api/x402/v1/fx-rate","summary":"Fiat currency exchange rate (ECB reference rates via Frankfurter, with ExchangeRate-API fallback for 150+ currencies), o","price":"$0.005","params":["from","to","amount","date"]},{"method":"GET","path":"/v1/ens-resolve","url":"https://scorptech.it.com/api/x402/v1/ens-resolve","summary":"Resolve an ENS name to its Ethereum address (plus optional text records like avatar, url, com.twitter), or reverse-resol","price":"$0.005","params":["name","address","texts"]},{"method":"GET","path":"/v1/geocode","url":"https://scorptech.it.com/api/x402/v1/geocode","summary":"Geocode an address or place name to latitude/longitude (US Census Geocoder for US street addresses, Photon/OpenStreetMap","price":"$0.005","params":["q","lat","lon","provider","limit","lang"]},{"method":"GET","path":"/v1/wallet-risk-basic","url":"https://scorptech.it.com/api/x402/v1/wallet-risk-basic","summary":"Basic heuristic risk screen for a Solana or EVM (Ethereum/Base/Arbitrum) wallet: activity & age, bundled OFAC-SDN / mixe","price":"$0.02","params":["address","chain","deep","lookback_blocks"]},{"method":"GET","path":"/v1/sanctions-screen","url":"https://scorptech.it.com/api/x402/v1/sanctions-screen","summary":"Sanctions screening of a person or company NAME against the official consolidated lists: US OFAC SDN + non-SDN, UK Sanct","price":"$0.01","params":["name","type","lists","minScore","limit"]},{"method":"GET","path":"/v1/sec-13f","url":"https://scorptech.it.com/api/x402/v1/sec-13f","summary":"Institutional holdings from a fund manager's latest SEC 13F-HR: the information table parsed from EDGAR, aggregated by C","price":"$0.02","params":["fund","limit","accession"]},{"method":"GET","path":"/v1/treasury-rates","url":"https://scorptech.it.com/api/x402/v1/treasury-rates","summary":"US Treasury rates: the daily par yield curve (1M-30Y, from treasury.gov) with day-over-day change in bp, 2s10s spread an","price":"$0.01","params":["date","curve","debt"]},{"method":"GET","path":"/v1/domain-availability","url":"https://scorptech.it.com/api/x402/v1/domain-availability","summary":"Bulk domain availability: for up to 20 domains, asks the authoritative registry over RDAP (server chosen from the IANA R","price":"$0.005","params":["domains"]},{"method":"GET","path":"/v1/eu-vat-validate","url":"https://scorptech.it.com/api/x402/v1/eu-vat-validate","summary":"Validate an EU VAT number live against the European Commission VIES service: valid/invalid, plus registered trader name ","price":"$0.01","params":["vat","country"]},{"method":"GET","path":"/v1/ip-reputation","url":"https://scorptech.it.com/api/x402/v1/ip-reputation","summary":"IP blocklist reputation for up to 50 IPv4/IPv6 addresses: membership in the Tor exit list, Spamhaus DROP/DROPv6, abuse.c","price":"$0.01","params":["ip"]},{"method":"GET","path":"/v1/cisa-kev","url":"https://scorptech.it.com/api/x402/v1/cisa-kev","summary":"CISA Known Exploited Vulnerabilities check: is a CVE (or up to 100) in CISA's KEV catalog, with date added, federal reme","price":"$0.01","params":["cve","vendor","product","days","ransomware","limit"]},{"method":"GET","path":"/v1/epss-score","url":"https://scorptech.it.com/api/x402/v1/epss-score","summary":"FIRST EPSS exploit-prediction scores for up to 100 CVEs: probability of exploitation in the next 30 days, percentile ran","price":"$0.01","params":["cve","date"]},{"method":"GET","path":"/v1/hsts-preload","url":"https://scorptech.it.com/api/x402/v1/hsts-preload","summary":"HSTS preload check for a domain: current status on the Chromium preload list (hstspreload.org), the official preloadable","price":"$0.01","params":["domain"]},{"method":"GET","path":"/v1/macro-indicators","url":"https://scorptech.it.com/api/x402/v1/macro-indicators","summary":"Macro indicators by country and year: GDP, real GDP growth, GDP per capita, inflation, unemployment, population, governm","price":"$0.01","params":["country","indicators","from","to","source"]},{"method":"GET","path":"/v1/osv-scan","url":"https://scorptech.it.com/api/x402/v1/osv-scan","summary":"Known vulnerabilities for ONE package version from OSV.dev (GitHub Advisory DB, PyPA, Go, RustSec, OSS-Fuzz, distro feed","price":"$0.02","params":["ecosystem","package","version","purl","details"]},{"method":"POST","path":"/v1/osv-scan","url":"https://scorptech.it.com/api/x402/v1/osv-scan","summary":"Dependency vulnerability scan (SCA) via OSV.dev querybatch: send up to 500 packages, or paste a lockfile (package-lock.j","price":"$0.02","params":[]},{"method":"GET","path":"/v1/iban-validate","url":"https://scorptech.it.com/api/x402/v1/iban-validate","summary":"Validate an IBAN offline: ISO 7064 mod-97 checksum, per-country length and BBAN structure from the SWIFT IBAN registry (","price":"$0.005","params":["iban","bic"]},{"method":"GET","path":"/v1/wikipedia","url":"https://scorptech.it.com/api/x402/v1/wikipedia","summary":"Wikipedia in one call: search any of 340+ language editions (titles, descriptions, snippets, thumbnails) and get the top","price":"$0.005","params":["q","title","lang","limit","summary"]},{"method":"GET","path":"/v1/pubmed-search","url":"https://scorptech.it.com/api/x402/v1/pubmed-search","summary":"Search PubMed biomedical literature via NCBI E-utilities (esearch + esummary + efetch): PMID, title, authors, journal, p","price":"$0.01","params":["query","limit","sort","from","to","abstracts"]},{"method":"GET","path":"/v1/feed-discover","url":"https://scorptech.it.com/api/x402/v1/feed-discover","summary":"Find a website's RSS/Atom/JSON feeds and read the latest items: parses <link rel=\"alternate\"> tags, likely feed links an","price":"$0.005","params":["url","items","probe"]},{"method":"GET","path":"/v1/evm-selector-decode","url":"https://scorptech.it.com/api/x402/v1/evm-selector-decode","summary":"Decode EVM function selectors, calldata and event topics: looks up 4-byte selectors (or 32-byte event topics) in 4byte.d","price":"$0.005","params":["selector","calldata","topic","signature","limit"]},{"method":"GET","path":"/v1/readability-extract","url":"https://scorptech.it.com/api/x402/v1/readability-extract","summary":"Extract the main article from a web page (Readability-style content scoring, no headless browser): clean body text with ","price":"$0.01","params":["url","maxChars","paragraphs"]},{"method":"GET","path":"/v1/openalex-search","url":"https://scorptech.it.com/api/x402/v1/openalex-search","summary":"Scholarly search over OpenAlex (250M+ works): works with title, authors + ORCID + institutions, year, venue, DOI, 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Give a balance, an annual percentage rate, and a fixed payment ","price":"$0.002","params":["balance","ratePct","payment","minPercent","minFloor","start"]},{"method":"POST","path":"/v1/invoice-total","url":"https://scorptech.it.com/api/x402/v1/invoice-total","summary":"Invoice math in exact cents. Each line is quantity × unit price, then an optional line discount, rounded half away from ","price":"$0.002","params":[]},{"method":"GET","path":"/v1/wage-calc","url":"https://scorptech.it.com/api/x402/v1/wage-calc","summary":"Gross pay in exact cents. Hourly: regular hours at the rate, overtime hours at a multiplier (default 1.5) and double-tim","price":"$0.001","params":["mode","rate","hours","regularHours","otHours","dtHours","otMultiplier","dtMultiplier","weeklyThreshold","periods"]},{"method":"GET","path":"/v1/prorate","url":"https://scorptech.it.com/api/x402/v1/prorate","summary":"Prorate a money amount across a date range, in exact cents. The window is half-open: from the start date up to but not i","price":"$0.001","params":["amount","periodStart","periodEnd","useStart","useEnd"]},{"method":"GET","path":"/v1/markup-margin","url":"https://scorptech.it.com/api/x402/v1/markup-margin","summary":"Price, margin and markup in exact cents. Analyze a cost and a price (margin is profit / price, markup is profit / cost).","price":"$0.001","params":["mode","cost","price","marginPct","markupPct","discounts"]},{"method":"GET","path":"/v1/text-width","url":"https://scorptech.it.com/api/x402/v1/text-width","summary":"Width of one line in Helvetica or Helvetica-Bold, in points. Glyph widths are the Adobe standard-14 AFM numbers in WinAn","price":"$0.001","params":["text","font","size"]},{"method":"POST","path":"/v1/text-wrap","url":"https://scorptech.it.com/api/x402/v1/text-wrap","summary":"Wrap text to a measure in Helvetica or Helvetica-Bold. Hard breaks are LF, CR, or CRLF. Inside a paragraph, spaces colla","price":"$0.001","params":[]},{"method":"GET","path":"/v1/text-ellipsis","url":"https://scorptech.it.com/api/x402/v1/text-ellipsis","summary":"Cut one Helvetica line so it fits a width, ending with an ellipsis (…) when the text is too long. The ellipsis is the Wi","price":"$0.001","params":["text","font","size","width"]},{"method":"GET","path":"/v1/text-fit","url":"https://scorptech.it.com/api/x402/v1/text-fit","summary":"Largest Helvetica size, on a fixed step, at which one line still fits a width. Sizes run from minSize to maxSize in step","price":"$0.001","params":["text","font","width","minSize","maxSize","step"]},{"method":"POST","path":"/v1/text-block","url":"https://scorptech.it.com/api/x402/v1/text-block","summary":"Wrap a Helvetica block and give its height. Line breaking is the same as text-wrap. Leading defaults to 1.2 times the fo","price":"$0.001","params":[]},{"method":"GET","path":"/v1/isbn-validate","url":"https://scorptech.it.com/api/x402/v1/isbn-validate","summary":"ISBN-10 and ISBN-13 check digit. Spaces and hyphens are ignored, and a 9-digit number is read as an ISBN-10 with a leadi","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/issn-validate","url":"https://scorptech.it.com/api/x402/v1/issn-validate","summary":"ISSN check digit (8 characters, the last may be X). Spaces and hyphens are ignored. The formatted form is the two groups","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/imei-validate","url":"https://scorptech.it.com/api/x402/v1/imei-validate","summary":"IMEI and IMEISV. A 15-digit IMEI must pass the Luhn check. A 14-digit IMEI has no check digit, and a 16-digit IMEISV has","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/ean-validate","url":"https://scorptech.it.com/api/x402/v1/ean-validate","summary":"GTIN check digit for EAN-8, UPC-A (12 digits), EAN-13 and GTIN-14. The weight alternates 3 and 1 from the right, startin","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/isin-validate","url":"https://scorptech.it.com/api/x402/v1/isin-validate","summary":"ISIN check digit (ISO 6166). Two letters, nine letters or digits, one check digit. Letters expand to their alphabet posi","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/cusip-validate","url":"https://scorptech.it.com/api/x402/v1/cusip-validate","summary":"CUSIP check digit. Nine characters from digits, letters, and * @ #. Spaces are ignored and hyphens are not. From the lef","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/figi-validate","url":"https://scorptech.it.com/api/x402/v1/figi-validate","summary":"FIGI check digit. Twelve characters with no vowels. The first two are letters and are not BS, BM, GG, GB or VG, and the ","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/lei-validate","url":"https://scorptech.it.com/api/x402/v1/lei-validate","summary":"LEI check digits, ISO 17442 with ISO 7064 mod 97-10. Twenty letters or digits, and the whole number is 1 mod 97. Spaces ","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/creditor-ref","url":"https://scorptech.it.com/api/x402/v1/creditor-ref","summary":"ISO 11649 creditor reference. RF, two check digits, then up to 21 letters or digits. Spaces, hyphens, dots, commas, slas","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/container-check","url":"https://scorptech.it.com/api/x402/v1/container-check","summary":"ISO 6346 container check digit. Four letters (the fourth is U, J, Z or R), six serial digits and a check digit. Letter v","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/cas-validate","url":"https://scorptech.it.com/api/x402/v1/cas-validate","summary":"CAS Registry Number check digit. Two to seven digits, a hyphen, two digits, a hyphen and a check digit. No leading zero.","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/meid-validate","url":"https://scorptech.it.com/api/x402/v1/meid-validate","summary":"MEID, the CDMA equipment id. Hex is 14 digits with an optional Luhn check in base 16. Decimal is 18 digits (manufacturer","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/resistor-code","url":"https://scorptech.it.com/api/x402/v1/resistor-code","summary":"IEC 60062 resistor color code. Decode 3 to 6 bands, or encode a resistance that lands exactly on a digit-and-multiplier ","price":"$0.001","params":["bands","ohms","count","tolerance","tempco"]},{"method":"GET","path":"/v1/ohm-law","url":"https://scorptech.it.com/api/x402/v1/ohm-law","summary":"Ohm's law for one passive resistor. Give any two of voltage (volts), current (amperes), resistance (ohms) and power (wat","price":"$0.001","params":["voltage","current","resistance","power"]},{"method":"GET","path":"/v1/voltage-divider","url":"https://scorptech.it.com/api/x402/v1/voltage-divider","summary":"Unloaded voltage divider. r1 is the top resistor, from the supply to the output. r2 is the bottom resistor, from the out","price":"$0.001","params":["vin","r1","r2"]},{"method":"GET","path":"/v1/resistor-network","url":"https://scorptech.it.com/api/x402/v1/resistor-network","summary":"Series or parallel equivalent of 2 to 24 resistances. Series adds them. Parallel is the reciprocal of the sum of the rec","price":"$0.001","params":["mode","values"]},{"method":"GET","path":"/v1/led-resistor","url":"https://scorptech.it.com/api/x402/v1/led-resistor","summary":"Series resistor for LEDs on one supply. Resistance is (supply − series × forward) / current, with volts and amperes. The","price":"$0.001","params":["supply","forward","current","series"]},{"method":"GET","path":"/v1/mac-address","url":"https://scorptech.it.com/api/x402/v1/mac-address","summary":"Normalize a MAC-48 or EUI-64. Accepts colon, hyphen, Cisco dotted (groups of four) or plain hex, any case. Returns lower","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/eui64","url":"https://scorptech.it.com/api/x402/v1/eui64","summary":"Modified EUI-64 from RFC 4291. A 48-bit MAC is expanded by flipping the U/L bit and inserting FF-FE in the middle. A 64-","price":"$0.001","params":["value"]},{"method":"GET","path":"/v1/slaac-address","url":"https://scorptech.it.com/api/x402/v1/slaac-address","summary":"IPv6 address a host builds for itself from a /64 prefix and a MAC (SLAAC, RFC 4862). The interface id is the modified EU","price":"$0.001","params":["prefix","mac"]},{"method":"GET","path":"/v1/solicited-node","url":"https://scorptech.it.com/api/x402/v1/solicited-node","summary":"IPv6 solicited-node multicast address, ff02::1:ffXX:XXXX, from the low 24 bits of an IPv6 address or from the last three","price":"$0.001","params":["address","mac"]},{"method":"GET","path":"/v1/mac-local","url":"https://scorptech.it.com/api/x402/v1/mac-local","summary":"A stable locally administered unicast MAC from a seed. SHA-256 of the seed, first six bytes, multicast bit cleared and t","price":"$0.001","params":["seed"]},{"method":"GET","path":"/v1/css-length","url":"https://scorptech.it.com/api/x402/v1/css-length","summary":"Convert one CSS absolute length. 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The grammar is GNU chmod: symbolic [ugoa]*([-+=][rwxXst]* or one of ugo), comma s","price":"$0.001","params":["mode","spec","umask","directory"]},{"method":"GET","path":"/v1/umask-mode","url":"https://scorptech.it.com/api/x402/v1/umask-mode","summary":"Mode of a file or directory created under a umask. A file starts from 666 and a directory from 777, then the umask bits ","price":"$0.001","params":["umask","kind"]},{"method":"GET","path":"/v1/permission-check","url":"https://scorptech.it.com/api/x402/v1/permission-check","summary":"Whether one class may read, write, or execute under a mode. 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The letter moves with the interval number, then the accidental is chosen to hit","price":"$0.001","params":["note","quality","number","dir"]},{"method":"GET","path":"/v1/music-scale","url":"https://scorptech.it.com/api/x402/v1/music-scale","summary":"Spell a scale on letters from a tonic. Major, minor, harmonic minor, melodic minor (the ascending form), the church mode","price":"$0.001","params":["note","mode"]},{"method":"GET","path":"/v1/music-chord","url":"https://scorptech.it.com/api/x402/v1/music-chord","summary":"Spell a chord up from a root: major, minor, diminished, augmented, sus2, sus4, dom7, maj7, min7, dim7, half-dim7, and si","price":"$0.001","params":["note","quality"]},{"method":"GET","path":"/v1/paper-size","url":"https://scorptech.it.com/api/x402/v1/paper-size","summary":"ISO 216 A, B and C sizes, plus the DL envelope, in millimeters. Portrait, so the width is the shorter side. 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An odd final byte is padded with zero. \"abcde\" i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/inet-checksum","url":"https://scorptech.it.com/api/x402/v1/inet-checksum","summary":"RFC 1071 internet checksum. Big-endian 16-bit words are added with end-around carry, then inverted. An odd final byte is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/byte-sum","url":"https://scorptech.it.com/api/x402/v1/byte-sum","summary":"The two historic Unix sums, matching GNU sum. bsd rotates the 16-bit sum right and counts 1024-byte blocks. sysv reduces","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cf-expand","url":"https://scorptech.it.com/api/x402/v1/cf-expand","summary":"Continued fraction of a non-negative rational, by the Euclidean algorithm. 13/8 is [1; 1, 1, 1, 2]. 355/113 is [3; 7, 16","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cf-value","url":"https://scorptech.it.com/api/x402/v1/cf-value","summary":"Exact value of a continued fraction [a0; a1, a2, ...]. Later terms are at least 1. [3; 7, 15, 1] and [3; 7, 16] are both","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cf-convergent","url":"https://scorptech.it.com/api/x402/v1/cf-convergent","summary":"One convergent of a continued fraction. Index 0 is a0/1. The last index is the value itself. 13/8 at index 3 is 5/3.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cf-best","url":"https://scorptech.it.com/api/x402/v1/cf-best","summary":"Best rational approximation with a denominator up to maxDen. It searches the convergents and the intermediate fractions.","price":"$0.001","params":[]},{"method":"GET","path":"/v1/cf-sqrt","url":"https://scorptech.it.com/api/x402/v1/cf-sqrt","summary":"Continued fraction of a square root. sqrt(2) is [1; 2, 2, 2, ...] and the period is 1. sqrt(3) has period 2. A perfect s","price":"$0.001","params":["s","count"]},{"method":"GET","path":"/v1/morse-encode","url":"https://scorptech.it.com/api/x402/v1/morse-encode","summary":"Encode text as international Morse (ITU-R M.1677). Letters are separated by one space and words by \" / \". SOS is ... ---","price":"$0.001","params":["text"]},{"method":"GET","path":"/v1/morse-decode","url":"https://scorptech.it.com/api/x402/v1/morse-decode","summary":"Decode international Morse. Letters are separated by one space and words by \" / \". The result is capitals. ... --- ... i","price":"$0.001","params":["code"]},{"method":"GET","path":"/v1/morse-timing","url":"https://scorptech.it.com/api/x402/v1/morse-timing","summary":"How long a Morse message takes at a given speed. The dit is 1200/wpm milliseconds. PARIS is 43 units, or 50 units once t","price":"$0.001","params":["text","wpm"]},{"method":"POST","path":"/v1/bezier-cubic","url":"https://scorptech.it.com/api/x402/v1/bezier-cubic","summary":"Point on a cubic Bezier at t, as an exact fraction. t is between 0 and 1. The arch (0,0), (0,1), (1,1), (1,0) at one hal","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bezier-quadratic","url":"https://scorptech.it.com/api/x402/v1/bezier-quadratic","summary":"Point on a quadratic Bezier at t, as an exact fraction. (0,0), (1,1), (2,0) at one half is (1, 1/2).","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bezier-split","url":"https://scorptech.it.com/api/x402/v1/bezier-split","summary":"Split a quadratic or cubic Bezier at t by de Casteljau. Left and right are the two new control polygons, and they meet a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bezier-derivative","url":"https://scorptech.it.com/api/x402/v1/bezier-derivative","summary":"Control points of the derivative of a quadratic or cubic Bezier. A cubic derivative is a quadratic, and each control is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bezier-tangent","url":"https://scorptech.it.com/api/x402/v1/bezier-tangent","summary":"Tangent vector of a quadratic or cubic Bezier at t. It is the derivative curve evaluated at t. The arch (0,0), (0,1), (1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/quat-mul","url":"https://scorptech.it.com/api/x402/v1/quat-mul","summary":"Hamilton product of two quaternions, scalar first. i times j is k, and j times i is minus k. Components are exact fracti","price":"$0.001","params":[]},{"method":"POST","path":"/v1/quat-norm","url":"https://scorptech.it.com/api/x402/v1/quat-norm","summary":"Conjugate and squared length of a quaternion. (1, 2, 3, 4) has squared length 30 and conjugate (1, -2, -3, -4). Unit mea","price":"$0.001","params":[]},{"method":"POST","path":"/v1/quat-inv","url":"https://scorptech.it.com/api/x402/v1/quat-inv","summary":"Inverse of a quaternion, the conjugate divided by the squared length. (1, 2, 3, 4) inverts to (1/30, -1/15, -1/10, -2/15","price":"$0.001","params":[]},{"method":"POST","path":"/v1/quat-rotate","url":"https://scorptech.it.com/api/x402/v1/quat-rotate","summary":"Rotate a vector by a quaternion, q v q inverse. k sends (1, 0, 0) to (-1, 0, 0), a half turn about z. Scaling q does not","price":"$0.001","params":[]},{"method":"POST","path":"/v1/quat-pow","url":"https://scorptech.it.com/api/x402/v1/quat-pow","summary":"Integer power of a quaternion. i squared is -1, and i to the fourth is 1. Exponent 0 is 1, including zero to the 0. A ne","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perm-sign","url":"https://scorptech.it.com/api/x402/v1/perm-sign","summary":"Sign of a permutation of 0..n-1, from the number of inversions. [2, 1, 0] has three inversions and sign -1. The identity","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perm-inverse","url":"https://scorptech.it.com/api/x402/v1/perm-inverse","summary":"Inverse of a permutation of 0..n-1. The inverse of [1, 2, 0] is [2, 0, 1]. Applying both is the identity.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perm-compose","url":"https://scorptech.it.com/api/x402/v1/perm-compose","summary":"Compose two permutations of the same length. a is applied first, then b, so the result at i is b[a[i]]. Both must use 0.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perm-next","url":"https://scorptech.it.com/api/x402/v1/perm-next","summary":"The next permutation in lexicographic order. After [0, 1, 2] comes [0, 2, 1]. The reverse, [2, 1, 0], is the last and ha","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perm-rank","url":"https://scorptech.it.com/api/x402/v1/perm-rank","summary":"Lexicographic rank of a permutation, or the permutation at a rank. [0, 1, 2] is rank 0 and [2, 0, 1] is rank 4. Give the","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pell","url":"https://scorptech.it.com/api/x402/v1/pell","summary":"The nth Pell number. P0 is 0, P1 is 1, and each later term is twice the previous one plus the one before that. An option","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pell-lucas","url":"https://scorptech.it.com/api/x402/v1/pell-lucas","summary":"The nth companion Pell-Lucas number. Q0 and Q1 are both 2, and later terms use the Pell recurrence. An optional modulus ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pell-equation","url":"https://scorptech.it.com/api/x402/v1/pell-equation","summary":"The nth solution of the Pell equation x^2 - 2y^2 = (-1)^n. x is half the companion Pell-Lucas number and y is the Pell n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pell-sum","url":"https://scorptech.it.com/api/x402/v1/pell-sum","summary":"The sum of Pell numbers, or companion Pell-Lucas numbers, from lo through hi inclusive. seq is p or q. lo must not pass ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pell-compose","url":"https://scorptech.it.com/api/x402/v1/pell-compose","summary":"Compose two solutions of x^2 - 2y^2 = ±1. The unit for index a multiplied by the unit for index b is the unit for a+b. a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/binom","url":"https://scorptech.it.com/api/x402/v1/binom","summary":"The binomial coefficient C(n, k). k must not exceed n, and both indexes run up to 4096. An optional modulus reduces the ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/catalan","url":"https://scorptech.it.com/api/x402/v1/catalan","summary":"The nth Catalan number. C0 and C1 are 1, C2 is 2, C3 is 5, and C_n = binom(2n, n) / (n+1). n runs from 0 to 2048. An opt","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ballot","url":"https://scorptech.it.com/api/x402/v1/ballot","summary":"The ballot number B(n, k) = (n-k+1)/(n+1) * binom(n+k, k), for 0 <= k <= n. B(n, n) is the nth Catalan number. k above n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/dyck-word","url":"https://scorptech.it.com/api/x402/v1/dyck-word","summary":"Check a Dyck word of parentheses and report its semilength and height. The empty word is valid. A prefix that goes negat","price":"$0.001","params":[]},{"method":"POST","path":"/v1/motzkin","url":"https://scorptech.it.com/api/x402/v1/motzkin","summary":"The nth Motzkin number. M0 and M1 are 1, M2 is 2, M4 is 9, from M_n = sum binom(n, 2k) C_k. n runs from 0 to 512. An opt","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mod-pow","url":"https://scorptech.it.com/api/x402/v1/mod-pow","summary":"base^exp mod m, exact, for a non-negative exponent. The modulus must be at least 1. A negative exponent is rejected. Dec","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mod-inv","url":"https://scorptech.it.com/api/x402/v1/mod-inv","summary":"The modular inverse of a modulo m, as a residue from 0 up to m-1. The modulus must be at least 1. A value that is not co","price":"$0.001","params":[]},{"method":"POST","path":"/v1/crt","url":"https://scorptech.it.com/api/x402/v1/crt","summary":"Solve simultaneous congruences x ≡ a_i (mod m_i). Shared factors are allowed when the system is consistent; an inconsist","price":"$0.001","params":[]},{"method":"POST","path":"/v1/jacobi","url":"https://scorptech.it.com/api/x402/v1/jacobi","summary":"The Jacobi symbol (a/n). n must be odd and positive. The value is -1, 0, or 1. An even or non-positive n is rejected.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/egcd","url":"https://scorptech.it.com/api/x402/v1/egcd","summary":"Extended gcd of a and b. Returns g, x, and y with g = a*x + b*y and g >= 0. gcd(0, 0) is rejected. Decimals have no lead","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pythag-gen","url":"https://scorptech.it.com/api/x402/v1/pythag-gen","summary":"The Pythagorean triple from m > n >= 1 and scale k >= 1. a is k(m^2-n^2), b is k(2mn), and c is k(m^2+n^2). primitive is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pythag-check","url":"https://scorptech.it.com/api/x402/v1/pythag-check","summary":"Check a^2 + b^2 = c^2 for positive integers, with c the hypotenuse. primitive is true when the three sides are coprime. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/heron-area","url":"https://scorptech.it.com/api/x402/v1/heron-area","summary":"Exact area data for a strict integer triangle. sixteen is 16 times the squared area. When sixteen is a perfect square, a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cosine-law","url":"https://scorptech.it.com/api/x402/v1/cosine-law","summary":"The cosine of the angle opposite side c, as a reduced rational (a^2+b^2-c^2)/(2ab). The sides must form a strict triangl","price":"$0.001","params":[]},{"method":"POST","path":"/v1/triangle-shape","url":"https://scorptech.it.com/api/x402/v1/triangle-shape","summary":"Classify a strict integer triangle. sides is equilateral, isosceles, or scalene. angles is acute, right, or obtuse, from","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stern","url":"https://scorptech.it.com/api/x402/v1/stern","summary":"Stern's diatomic sequence. s(0) is 0, s(1) is 1, even terms repeat s(n/2), and an odd term is the sum of its two neighbo","price":"$0.001","params":[]},{"method":"POST","path":"/v1/calkin-wilf","url":"https://scorptech.it.com/api/x402/v1/calkin-wilf","summary":"The Calkin-Wilf tree. Give n to get the nth positive rational, starting at 1/1, or give a reduced num and den to get its","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mediant","url":"https://scorptech.it.com/api/x402/v1/mediant","summary":"The mediant of a/b and c/d, which is (a+c)/(b+d) reduced by their gcd. Numerators may be zero. A non-positive denominato","price":"$0.001","params":[]},{"method":"POST","path":"/v1/farey-adjacent","url":"https://scorptech.it.com/api/x402/v1/farey-adjacent","summary":"Whether a/b and c/d are Farey neighbors. They are adjacent when the determinant ad-bc is 1 or -1. A non-positive denomin","price":"$0.001","params":[]},{"method":"POST","path":"/v1/farey-child","url":"https://scorptech.it.com/api/x402/v1/farey-child","summary":"The mediant child of a Farey-neighbor pair. a/b and c/d must satisfy |ad-bc| = 1. A pair that is not adjacent is rejecte","price":"$0.001","params":[]},{"method":"POST","path":"/v1/partition","url":"https://scorptech.it.com/api/x402/v1/partition","summary":"The number of integer partitions p(n), from the pentagonal recurrence. p(0) is 1, p(5) is 7, and p(10) is 42. n runs fro","price":"$0.001","params":[]},{"method":"POST","path":"/v1/partition-distinct","url":"https://scorptech.it.com/api/x402/v1/partition-distinct","summary":"The number of partitions of n into distinct positive parts. This equals the number of partitions into odd parts. q(0) is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pentagonal","url":"https://scorptech.it.com/api/x402/v1/pentagonal","summary":"The generalized pentagonal number n(3n-1)/2. n may be negative. n = 1 gives 1, n = -1 gives 2, and n = 0 is rejected. Pa","price":"$0.001","params":[]},{"method":"POST","path":"/v1/partition-conjugate","url":"https://scorptech.it.com/api/x402/v1/partition-conjugate","summary":"The conjugate of an integer partition, read down the columns of its Ferrers diagram. Parts must be positive and nonincre","price":"$0.001","params":[]},{"method":"POST","path":"/v1/partition-parts","url":"https://scorptech.it.com/api/x402/v1/partition-parts","summary":"Check an integer partition and return its sum and length. Parts must be positive and nonincreasing. The empty list is th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/base-convert","url":"https://scorptech.it.com/api/x402/v1/base-convert","summary":"Convert an integer from one base to another. Bases run from 2 to 36, and letters may be any case. A digit that is not va","price":"$0.001","params":[]},{"method":"POST","path":"/v1/digit-sum","url":"https://scorptech.it.com/api/x402/v1/digit-sum","summary":"The sum of the decimal digits of an integer, ignoring the sign. 199 sums to 19. The decimal has no leading zeros. A malf","price":"$0.001","params":[]},{"method":"POST","path":"/v1/digital-root","url":"https://scorptech.it.com/api/x402/v1/digital-root","summary":"The digital root of a decimal integer. Zero stays zero, and any other magnitude is 1 + (n-1) mod 9, so multiples of 9 ha","price":"$0.001","params":[]},{"method":"POST","path":"/v1/digit-rev","url":"https://scorptech.it.com/api/x402/v1/digit-rev","summary":"Reverse the decimal digits of an integer. Leading zeros of the reversal are dropped, so 1200 becomes 21. The sign is kep","price":"$0.001","params":[]},{"method":"POST","path":"/v1/repunit","url":"https://scorptech.it.com/api/x402/v1/repunit","summary":"The repunit of length n, (10^n - 1)/9. R1 is 1 and R2 is 11. n = 0 is rejected. n runs up to 500. Past 15 digits the val","price":"$0.001","params":[]},{"method":"POST","path":"/v1/gray-encode","url":"https://scorptech.it.com/api/x402/v1/gray-encode","summary":"Encode a non-negative integer as a binary reflected Gray code. The code is n xor (n shifted right by one). Zero stays ze","price":"$0.001","params":[]},{"method":"POST","path":"/v1/gray-decode","url":"https://scorptech.it.com/api/x402/v1/gray-decode","summary":"Decode a binary reflected Gray code back to its index. The walk xors the code with each of its right shifts. A negative ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/gray-next","url":"https://scorptech.it.com/api/x402/v1/gray-next","summary":"The next binary reflected Gray code after g. It is the code of the index one past the index of g. A negative code is rej","price":"$0.001","params":[]},{"method":"POST","path":"/v1/gray-flip","url":"https://scorptech.it.com/api/x402/v1/gray-flip","summary":"The single bit that flips in the binary reflected Gray code when the index goes from n to n+1. Bit 0 is the least signif","price":"$0.001","params":[]},{"method":"POST","path":"/v1/gray-bits","url":"https://scorptech.it.com/api/x402/v1/gray-bits","summary":"The binary reflected Gray code of n, written in exactly width bits. Bit 0 is on the right. n must be smaller than 2^widt","price":"$0.001","params":[]},{"method":"POST","path":"/v1/poly-add","url":"https://scorptech.it.com/api/x402/v1/poly-add","summary":"Add two integer polynomials, lowest degree first. Trailing zeros are dropped, and the zero polynomial is an empty list. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/poly-mul","url":"https://scorptech.it.com/api/x402/v1/poly-mul","summary":"Multiply two integer polynomials, lowest degree first. The zero polynomial is an empty list. An optional modulus reduces","price":"$0.001","params":[]},{"method":"POST","path":"/v1/poly-eval","url":"https://scorptech.it.com/api/x402/v1/poly-eval","summary":"Evaluate an integer polynomial at an integer point, by Horner. The zero polynomial evaluates to 0. An optional modulus r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/poly-deriv","url":"https://scorptech.it.com/api/x402/v1/poly-deriv","summary":"Differentiate an integer polynomial. The derivative of a constant is the zero polynomial, which is an empty list. An opt","price":"$0.001","params":[]},{"method":"POST","path":"/v1/poly-div","url":"https://scorptech.it.com/api/x402/v1/poly-div","summary":"Exact quotient of two integer polynomials. The divisor must be non-zero, and every leading step must divide in the integ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/totient","url":"https://scorptech.it.com/api/x402/v1/totient","summary":"Euler totient phi(n), the count of integers from 1 to n that are coprime to n. phi(1) is 1. n = 0 is rejected. n runs up","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mobius","url":"https://scorptech.it.com/api/x402/v1/mobius","summary":"The Mobius function mu(n). It is 0 when n has a squared prime factor, and otherwise (-1)^k for k distinct primes. mu(1) ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/divisor-count","url":"https://scorptech.it.com/api/x402/v1/divisor-count","summary":"The number of positive divisors of n. A prime has 2, and d(1) is 1. n = 0 is rejected. n runs up to 10^12.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/divisor-sum","url":"https://scorptech.it.com/api/x402/v1/divisor-sum","summary":"The sum of the positive divisors of n. sigma(6) is 12 and sigma(28) is 56. n = 0 is rejected. A value past 15 digits is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prime-check","url":"https://scorptech.it.com/api/x402/v1/prime-check","summary":"Whether n is prime, by trial division. 1 is not prime. n < 1 is rejected. n runs up to 10^12.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/egypt-greedy","url":"https://scorptech.it.com/api/x402/v1/egypt-greedy","summary":"The greedy Egyptian fraction for a proper positive fraction a/b. 5/6 is 1/2 + 1/3. A fraction that is not strictly betwe","price":"$0.001","params":[]},{"method":"POST","path":"/v1/unit-sum","url":"https://scorptech.it.com/api/x402/v1/unit-sum","summary":"The reduced sum of unit fractions 1/d. Dens must be positive. An empty list or a non-positive denominator is rejected.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/harmonic","url":"https://scorptech.it.com/api/x402/v1/harmonic","summary":"The nth harmonic number as a reduced fraction. H1 is 1/1, H2 is 3/2, and H6 is 49/20. n < 1 is rejected. n runs up to 20","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sylvester","url":"https://scorptech.it.com/api/x402/v1/sylvester","summary":"The nth Sylvester number. s(0) is 2, and each later term is s*(s-1)+1. A negative index is rejected. n runs up to 8. Pas","price":"$0.001","params":[]},{"method":"POST","path":"/v1/egypt-check","url":"https://scorptech.it.com/api/x402/v1/egypt-check","summary":"Check that a sum of unit fractions equals the fraction a/b. A match returns the reduced sum. A sum that differs, or a no","price":"$0.001","params":[]},{"method":"POST","path":"/v1/jacobsthal","url":"https://scorptech.it.com/api/x402/v1/jacobsthal","summary":"The nth Jacobsthal number. J0 is 0, J1 is 1, and each later term is the previous one plus twice the one before that. An ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/jacobsthal-lucas","url":"https://scorptech.it.com/api/x402/v1/jacobsthal-lucas","summary":"The nth Jacobsthal-Lucas number. L0 is 2, L1 is 1, and later terms use the Jacobsthal recurrence. An optional modulus re","price":"$0.001","params":[]},{"method":"POST","path":"/v1/jacobsthal-sum","url":"https://scorptech.it.com/api/x402/v1/jacobsthal-sum","summary":"The sum of Jacobsthal numbers, or Jacobsthal-Lucas numbers, from lo through hi inclusive. seq is j or l. lo must not pas","price":"$0.001","params":[]},{"method":"POST","path":"/v1/jacobsthal-add","url":"https://scorptech.it.com/api/x402/v1/jacobsthal-add","summary":"The Jacobsthal addition rule. (L(a)*J(b) + J(a)*L(b)) / 2 is J(a+b). a+b must be at most 8192.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/jacobsthal-power","url":"https://scorptech.it.com/api/x402/v1/jacobsthal-power","summary":"The power identity J(n) + J(n+1) = 2^n. left is J(n), right is J(n+1), and value is their sum. n runs from 0 to 8191 so ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/padovan","url":"https://scorptech.it.com/api/x402/v1/padovan","summary":"The nth Padovan number. P0, P1, and P2 are 1, and each later term is the sum of the term two back and the term three bac","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perrin","url":"https://scorptech.it.com/api/x402/v1/perrin","summary":"The nth Perrin number. R0 is 3, R1 is 0, R2 is 2, and later terms use the Padovan recurrence. An optional modulus reduce","price":"$0.001","params":[]},{"method":"POST","path":"/v1/padovan-sum","url":"https://scorptech.it.com/api/x402/v1/padovan-sum","summary":"The sum of Padovan numbers, or Perrin numbers, from lo through hi inclusive. seq is p or r. lo must not pass hi. Each in","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perrin-divide","url":"https://scorptech.it.com/api/x402/v1/perrin-divide","summary":"The remainder of the nth Perrin number divided by n. A prime n divides R(n), so the remainder is 0. n runs from 1 to 819","price":"$0.001","params":[]},{"method":"POST","path":"/v1/padovan-step","url":"https://scorptech.it.com/api/x402/v1/padovan-step","summary":"The Padovan step identity P(n) = P(n-1) + P(n-5). left is P(n-1) and right is P(n-5). n runs from 5 to 8192. Past 15 dig","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stirling2","url":"https://scorptech.it.com/api/x402/v1/stirling2","summary":"The Stirling number of the second kind S(n,k), the number of ways to partition n objects into k nonempty unlabeled sets.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bell","url":"https://scorptech.it.com/api/x402/v1/bell","summary":"The nth Bell number, the number of partitions of a set of n objects. B(0) and B(1) are 1, and B(5) is 52. n runs from 0 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stirling2-row","url":"https://scorptech.it.com/api/x402/v1/stirling2-row","summary":"Row n of Stirling numbers of the second kind, from S(n,0) through S(n,n), plus the Bell number that sums the row. n runs","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bell-sum","url":"https://scorptech.it.com/api/x402/v1/bell-sum","summary":"The sum of Bell numbers from lo through hi inclusive. lo must not pass hi. Each index runs from 0 to 80. Past 15 digits ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stirling2-rec","url":"https://scorptech.it.com/api/x402/v1/stirling2-rec","summary":"The Stirling recurrence S(n,k) = k*S(n-1,k) + S(n-1,k-1). term is the first summand and diag is the second. n and k run ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/delannoy","url":"https://scorptech.it.com/api/x402/v1/delannoy","summary":"The Delannoy number D(m,n), the number of paths from (0,0) to (m,n) using steps (1,0), (0,1), and (1,1). D(3,5) is 231. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/delannoy-central","url":"https://scorptech.it.com/api/x402/v1/delannoy-central","summary":"The central Delannoy number D(n,n). D(0,0) is 1 and D(3,3) is 63. n runs from 0 to 64. Past 15 digits the value is a dec","price":"$0.001","params":[]},{"method":"POST","path":"/v1/schroder","url":"https://scorptech.it.com/api/x402/v1/schroder","summary":"The nth large Schröder number. S(0) is 1, S(1) is 2, and S(5) is 394. n runs from 0 to 64. Past 15 digits the value is a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/schroder-little","url":"https://scorptech.it.com/api/x402/v1/schroder-little","summary":"The nth little Schröder number. It is 1 at n = 0 and half the large Schröder number for n > 0. s(5) is 197. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/delannoy-sum","url":"https://scorptech.it.com/api/x402/v1/delannoy-sum","summary":"The sum of Delannoy numbers D(m,0) through D(m,hi). The sum through D(4,4) is 501. m and hi run from 0 to 64. Past 15 di","price":"$0.001","params":[]},{"method":"POST","path":"/v1/narayana","url":"https://scorptech.it.com/api/x402/v1/narayana","summary":"The Narayana number N(n,k). N(5,3) is 20. A k outside 1..n yields 0. n runs from 1 to 60 and k from 0 to 60. Past 15 dig","price":"$0.001","params":[]},{"method":"POST","path":"/v1/narayana-row","url":"https://scorptech.it.com/api/x402/v1/narayana-row","summary":"Narayana numbers N(n,1) through N(n,n). Their sum is the Catalan number C(n). The row for n = 4 is 1, 6, 6, 1 and sums t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/narayana-check","url":"https://scorptech.it.com/api/x402/v1/narayana-check","summary":"The Narayana product C(n,k)*C(n,k-1). It is exactly n times N(n,k). product is that binomial product and value is the Na","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fuss-catalan","url":"https://scorptech.it.com/api/x402/v1/fuss-catalan","summary":"The Fuss-Catalan number C(p*n+1, n) / (p*n+1). FC(2,n) is the Catalan number, so FC(2,5) is 42. p runs from 1 to 8 and n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fuss-check","url":"https://scorptech.it.com/api/x402/v1/fuss-check","summary":"The Fuss-Catalan division. num is C(p*n+1, n), den is p*n+1, and value is the exact quotient. p runs from 1 to 8 and n f","price":"$0.001","params":[]},{"method":"POST","path":"/v1/chebyshev-t","url":"https://scorptech.it.com/api/x402/v1/chebyshev-t","summary":"The Chebyshev polynomial of the first kind T(n,x) at an integer x. T(0,x) is 1, T(1,x) is x, and T(2,3) is 17. n runs fr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/chebyshev-u","url":"https://scorptech.it.com/api/x402/v1/chebyshev-u","summary":"The Chebyshev polynomial of the second kind U(n,x) at an integer x. U(0,x) is 1, U(1,x) is 2*x, and U(3,2) is 56. n runs","price":"$0.001","params":[]},{"method":"POST","path":"/v1/chebyshev-disc","url":"https://scorptech.it.com/api/x402/v1/chebyshev-disc","summary":"The Chebyshev discriminant identity T(n,x)^2 - (x^2-1)*U(n-1,x)^2 = 1. gap is that difference. n runs from 1 to 64. Past","price":"$0.001","params":[]},{"method":"POST","path":"/v1/chebyshev-prod","url":"https://scorptech.it.com/api/x402/v1/chebyshev-prod","summary":"The product identity 2*T(a,x)*T(b,x) = T(a+b,x) + T(|a-b|,x). twice is the left side and sum is the right side. a+b must","price":"$0.001","params":[]},{"method":"POST","path":"/v1/chebyshev-rec","url":"https://scorptech.it.com/api/x402/v1/chebyshev-rec","summary":"One step of the Chebyshev recurrence. kind is t or u. prev is degree n-1, older is degree n-2, and value is 2*x*prev - o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/tribonacci","url":"https://scorptech.it.com/api/x402/v1/tribonacci","summary":"The nth Tribonacci number. T0 and T1 are 0, T2 is 1, and each later term is the sum of the previous three. An optional m","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trib-sum","url":"https://scorptech.it.com/api/x402/v1/trib-sum","summary":"The sum of Tribonacci numbers from lo through hi inclusive. The sum from T0 through T10 is 177. lo must not pass hi. Eac","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trib-step","url":"https://scorptech.it.com/api/x402/v1/trib-step","summary":"The Tribonacci step T(n) = T(n-1) + T(n-2) + T(n-3). left is T(n-3), mid is T(n-2), and right is T(n-1). n runs from 3 t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trib-index","url":"https://scorptech.it.com/api/x402/v1/trib-index","summary":"The smallest index n with T(n) equal to value. 0 is index 0, and 1 is index 2. A number that is not a Tribonacci term, o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trib-mod","url":"https://scorptech.it.com/api/x402/v1/trib-mod","summary":"T(n) mod m. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 to 4096. The residue is small","price":"$0.001","params":[]},{"method":"POST","path":"/v1/thue-morse","url":"https://scorptech.it.com/api/x402/v1/thue-morse","summary":"The nth Thue-Morse bit, the parity of the number of 1-bits in n. t(0) is 0, t(1) is 1, and t(2) is 1. n runs from 0 to 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/thue-sum","url":"https://scorptech.it.com/api/x402/v1/thue-sum","summary":"The number of 1-bits in the Thue-Morse sequence from lo through hi inclusive. The sum from t(0) through t(7) is 4. lo mu","price":"$0.001","params":[]},{"method":"POST","path":"/v1/thue-prefix","url":"https://scorptech.it.com/api/x402/v1/thue-prefix","summary":"The first n Thue-Morse bits as a string of 0 and 1. The first 8 bits are 01101001. n runs from 1 to 4096.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/thue-cube","url":"https://scorptech.it.com/api/x402/v1/thue-cube","summary":"Find a cube in a 0/1 string, a block xxx. 000 is a cube at index 0 of length 1. A Thue-Morse prefix has no cube. An empt","price":"$0.001","params":[]},{"method":"POST","path":"/v1/thue-weight","url":"https://scorptech.it.com/api/x402/v1/thue-weight","summary":"The number of 1s in t(0) through t(n), and the bit t(n) itself. Through n = 7 the weight is 4. n runs from 0 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/rudin-shapiro","url":"https://scorptech.it.com/api/x402/v1/rudin-shapiro","summary":"The Rudin-Shapiro bit of n, the parity of the number of overlapping 11 blocks in its binary expansion. n = 3 is 11, so t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/baum-sweet","url":"https://scorptech.it.com/api/x402/v1/baum-sweet","summary":"The Baum-Sweet bit of n. It is 1 when every run of 0s in the binary expansion has even length, and 0 when some run of 0s","price":"$0.001","params":[]},{"method":"POST","path":"/v1/paperfold","url":"https://scorptech.it.com/api/x402/v1/paperfold","summary":"The regular paperfolding bit of n. Write n = m * 2^k with m odd. The bit is 1 when m is 1 mod 4, and 0 when m is 3 mod 4","price":"$0.001","params":[]},{"method":"POST","path":"/v1/golay-rudin","url":"https://scorptech.it.com/api/x402/v1/golay-rudin","summary":"The Golay-Rudin-Shapiro partial sum. Each term is +1 or -1 as the Rudin-Shapiro sign of k, and the value is the sum from","price":"$0.001","params":[]},{"method":"POST","path":"/v1/morse-run","url":"https://scorptech.it.com/api/x402/v1/morse-run","summary":"The Thue-Morse run that contains n. The bit is the parity of the 1-bits of n. A run has length 1 or 2. Index 0 is a run ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucas","url":"https://scorptech.it.com/api/x402/v1/lucas","summary":"The nth Lucas number. L0 is 2, L1 is 1, and each later term is the sum of the previous two. L5 is 11 and L10 is 123. An ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucas-sum","url":"https://scorptech.it.com/api/x402/v1/lucas-sum","summary":"The sum of Lucas numbers from lo through hi inclusive. The sum from L0 through L5 is 28. The sum equals L(hi+2) - L(lo+1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucas-index","url":"https://scorptech.it.com/api/x402/v1/lucas-index","summary":"The smallest index n with L(n) equal to value. 2 is index 0, and 1 is index 1. A number that is not a Lucas number, or a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucas-mod","url":"https://scorptech.it.com/api/x402/v1/lucas-mod","summary":"L(n) mod m. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 to 4096. The residue is small","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucas-disc","url":"https://scorptech.it.com/api/x402/v1/lucas-disc","summary":"The Lucas discriminant L(n)^2 - 5*F(n)^2, which equals 4*(-1)^n. F is the Fibonacci sequence with F0 = 0 and F1 = 1. For","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ruler","url":"https://scorptech.it.com/api/x402/v1/ruler","summary":"The ruler value of n, the exponent of 2 in n. v2(1) is 0, v2(8) is 3, and v2(12) is 2. n runs from 1 to 1000000000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ruler-sum","url":"https://scorptech.it.com/api/x402/v1/ruler-sum","summary":"The sum of the ruler values v2(1) through v2(n). It equals n minus the number of 1-bits in n. The sum through 15 is 11. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ruler-prefix","url":"https://scorptech.it.com/api/x402/v1/ruler-prefix","summary":"The ruler values v2(1) through v2(n), written as comma-separated integers. The first 8 values are 0,1,0,2,0,1,0,3. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ruler-max","url":"https://scorptech.it.com/api/x402/v1/ruler-max","summary":"The largest ruler value on 1 through n, and the smallest index where it occurs. That index is the greatest power of 2 th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ruler-odd","url":"https://scorptech.it.com/api/x402/v1/ruler-odd","summary":"Remove every factor of 2 from n. The odd part of 12 is 3 and the valuation is 2. The odd part of 7 is 7. n runs from 1 t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-step","url":"https://scorptech.it.com/api/x402/v1/collatz-step","summary":"One Collatz step. An even n is halved and an odd n becomes 3n+1. 6 goes to 3, and 1 goes to 4. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-length","url":"https://scorptech.it.com/api/x402/v1/collatz-length","summary":"The number of Collatz steps from n until 1. The length of 1 is 0, the length of 6 is 8, and the length of 27 is 111. n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-peak","url":"https://scorptech.it.com/api/x402/v1/collatz-peak","summary":"The largest value on the Collatz trajectory from n through 1, including n. The peak of 1 is 1, the peak of 6 is 16, and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-parity","url":"https://scorptech.it.com/api/x402/v1/collatz-parity","summary":"The parity of each term on the Collatz trajectory from n through 1. 0 is even and 1 is odd. The parity of 6 is 010100001","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-glide","url":"https://scorptech.it.com/api/x402/v1/collatz-glide","summary":"The Collatz glide of n, the number of steps until a value strictly below n. The glide of 1 is 0, the glide of 6 is 1, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wythoff","url":"https://scorptech.it.com/api/x402/v1/wythoff","summary":"The nth lower Wythoff number, floor(n*phi), where phi is (1+sqrt(5))/2. A(1) is 1, A(2) is 3, and A(5) is 8. n runs from","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wythoff-upper","url":"https://scorptech.it.com/api/x402/v1/wythoff-upper","summary":"The nth upper Wythoff number, floor(n*phi^2), which is A(n)+n. B(1) is 2, B(2) is 5, and B(5) is 13. n runs from 1 to 10","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wythoff-pair","url":"https://scorptech.it.com/api/x402/v1/wythoff-pair","summary":"The nth Wythoff pair. lower is floor(n*phi) and upper is lower+n. The pair for n = 1 is 1 and 2. The pair for n = 5 is 8","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wythoff-inverse","url":"https://scorptech.it.com/api/x402/v1/wythoff-inverse","summary":"Which Wythoff sequence contains m, and at which index. 1 is the lower number at index 1. 2 is the upper number at index ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/beatty","url":"https://scorptech.it.com/api/x402/v1/beatty","summary":"The Beatty term floor(n*sqrt(d)). floor(5*sqrt(2)) is 7, and floor(2*sqrt(3)) is 3. n runs from 0 to 1000000 and d runs ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kolakoski","url":"https://scorptech.it.com/api/x402/v1/kolakoski","summary":"The nth Kolakoski term, using a 1-based index. The sequence is 1s and 2s, and it starts 1, 2, 2, 1, 1. Term 7 is 1. n ru","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kolakoski-prefix","url":"https://scorptech.it.com/api/x402/v1/kolakoski-prefix","summary":"The first n Kolakoski terms as a string of the digits 1 and 2. The first 10 terms are 1221121221. n runs from 1 to 4096.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kolakoski-sum","url":"https://scorptech.it.com/api/x402/v1/kolakoski-sum","summary":"The sum of the first n Kolakoski terms. The sum of the first 10 terms is 15. n runs from 1 to 100000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kolakoski-run","url":"https://scorptech.it.com/api/x402/v1/kolakoski-run","summary":"The Kolakoski run that contains the 1-based index n. A run has length 1 or 2. Index 1 is a single 1. Index 2 starts two ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kolakoski-check","url":"https://scorptech.it.com/api/x402/v1/kolakoski-check","summary":"Whether a string of the digits 1 and 2 is a prefix of the Kolakoski sequence. 122112 is a prefix. An empty string, or an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/carmichael","url":"https://scorptech.it.com/api/x402/v1/carmichael","summary":"Carmichael's lambda of n, the exponent of the multiplicative group mod n. lambda(1) is 1, lambda(8) is 2, lambda(p) is p","price":"$0.001","params":[]},{"method":"POST","path":"/v1/jordan-totient","url":"https://scorptech.it.com/api/x402/v1/jordan-totient","summary":"The Jordan totient J(k, n). J(1, n) is Euler's totient. J(1, 6) is 2 and J(2, 6) is 24. k runs from 1 to 12 and n runs f","price":"$0.001","params":[]},{"method":"POST","path":"/v1/dedekind-psi","url":"https://scorptech.it.com/api/x402/v1/dedekind-psi","summary":"Dedekind's psi function, n times the product of (1+1/p) over the distinct prime factors of n. psi(1) is 1, psi(2) is 3, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/aliquot","url":"https://scorptech.it.com/api/x402/v1/aliquot","summary":"The aliquot part of n, the sum of proper divisors. The aliquot part of 1 is 0, of 9 is 4, and of 6 is 6. n runs from 1 t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perfect-check","url":"https://scorptech.it.com/api/x402/v1/perfect-check","summary":"Classify n from its divisor sum. 6 and 28 are perfect. 12 is abundant. 8 is deficient. The class is perfect, abundant, o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hermite","url":"https://scorptech.it.com/api/x402/v1/hermite","summary":"The physicist Hermite polynomial H(n, x) at an integer x. H(0, x) is 1, H(1, x) is 2x, H(2, 1) is 2, and H(3, 1) is -4. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hermite-prob","url":"https://scorptech.it.com/api/x402/v1/hermite-prob","summary":"The probabilist Hermite polynomial He(n, x) at an integer x. He(0, x) is 1, He(1, x) is x, and He(2, 1) is 0. n runs fro","price":"$0.001","params":[]},{"method":"POST","path":"/v1/laguerre","url":"https://scorptech.it.com/api/x402/v1/laguerre","summary":"The Laguerre polynomial L(n, x) at an integer x, as a reduced fraction. L(n, 0) is 1, L(1, 1) is 0, and L(2, 1) is -1/2.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/legendre-pol","url":"https://scorptech.it.com/api/x402/v1/legendre-pol","summary":"The Legendre polynomial P(n, x) at an integer x, as a reduced fraction. P(n, 1) is 1, and P(2, 2) is 11/2. n runs from 0","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hermite-deriv","url":"https://scorptech.it.com/api/x402/v1/hermite-deriv","summary":"The derivative of the physicist Hermite polynomial, H'(n, x) = 2*n*H(n-1, x). At n = 3 and x = 1 the derivative is 12. n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stirling1","url":"https://scorptech.it.com/api/x402/v1/stirling1","summary":"The signed Stirling number of the first kind s(n, k). s(4, 2) is 11 and s(5, 2) is -50. s(n, 0) is 0 for n > 0. n runs f","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stirling1-unsigned","url":"https://scorptech.it.com/api/x402/v1/stirling1-unsigned","summary":"The unsigned Stirling number of the first kind, the absolute value of s(n, k). It counts permutations of n with k cycles","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stirling1-row","url":"https://scorptech.it.com/api/x402/v1/stirling1-row","summary":"Row n of the signed Stirling numbers of the first kind, from s(n, 0) through s(n, n). The absolute values sum to n!. Row","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stirling1-rec","url":"https://scorptech.it.com/api/x402/v1/stirling1-rec","summary":"One step of s(n, k) = s(n-1, k-1) - (n-1)*s(n-1, k). For n = 5 and k = 2, the previous term is -6, the scaled term is 44","price":"$0.001","params":[]},{"method":"POST","path":"/v1/subfactorial","url":"https://scorptech.it.com/api/x402/v1/subfactorial","summary":"The subfactorial !n, the number of derangements of n objects. !0 is 1, !1 is 0, !2 is 1, and !5 is 44. n runs from 0 to ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zeckendorf","url":"https://scorptech.it.com/api/x402/v1/zeckendorf","summary":"The Zeckendorf representation of n. Fibonacci numbers start F1 = 1, F2 = 1, F3 = 2, and no two used indices are consecut","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zeckendorf-sum","url":"https://scorptech.it.com/api/x402/v1/zeckendorf-sum","summary":"Sum Fibonacci numbers by index, using F1 = 1 and F2 = 1. Indices must be at least 2 and no two may be consecutive. Indic","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zeckendorf-bits","url":"https://scorptech.it.com/api/x402/v1/zeckendorf-bits","summary":"The Zeckendorf bit string of n, from the largest used Fibonacci index down to F2. The bit string of 1 is 1, of 4 is 101,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fibword","url":"https://scorptech.it.com/api/x402/v1/fibword","summary":"The nth bit of the infinite Fibonacci word, starting at n = 0. The word is the fixed point of 0 to 01 and 1 to 0, and it","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fibword-prefix","url":"https://scorptech.it.com/api/x402/v1/fibword-prefix","summary":"The first n bits of the Fibonacci word. The first 8 bits are 01001010, and the first 16 bits are 0100101001001010. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/little-omega","url":"https://scorptech.it.com/api/x402/v1/little-omega","summary":"The number of distinct prime factors of n. omega(1) is 0, omega(4) is 1, and omega(30) is 3. n runs from 1 to 1000000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/big-omega","url":"https://scorptech.it.com/api/x402/v1/big-omega","summary":"The number of prime factors of n counted with multiplicity. Omega(1) is 0, Omega(4) is 2, and Omega(12) is 3. n runs fro","price":"$0.001","params":[]},{"method":"POST","path":"/v1/radical","url":"https://scorptech.it.com/api/x402/v1/radical","summary":"The radical of n, the product of its distinct prime factors. rad(1) is 1, rad(12) is 6, and rad(36) is 6. n runs from 1 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sopfr","url":"https://scorptech.it.com/api/x402/v1/sopfr","summary":"The sum of prime factors of n with multiplicity. sopfr(1) is 0, sopfr(12) is 7, and sopfr(36) is 10. n runs from 1 to 10","price":"$0.001","params":[]},{"method":"POST","path":"/v1/squarefree","url":"https://scorptech.it.com/api/x402/v1/squarefree","summary":"Whether n is squarefree, together with its radical. 1 and 30 are squarefree. 4 and 12 are not. n runs from 1 to 10000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bernoulli","url":"https://scorptech.it.com/api/x402/v1/bernoulli","summary":"The nth Bernoulli number as a reduced fraction, from the Akiyama-Tanigawa algorithm. B0 is 1, B1 is 1/2, B2 is 1/6, and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bernoulli-denom","url":"https://scorptech.it.com/api/x402/v1/bernoulli-denom","summary":"The von Staudt denominator of B(m) for even m at least 2: the product of primes p such that p-1 divides m. The denominat","price":"$0.001","params":[]},{"method":"POST","path":"/v1/von-staudt","url":"https://scorptech.it.com/api/x402/v1/von-staudt","summary":"The von Staudt-Clausen integer. For even m at least 2, B(m) plus the sum of 1/p over primes p with p-1 dividing m is an ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/tangent-number","url":"https://scorptech.it.com/api/x402/v1/tangent-number","summary":"The nth tangent number. T1 is 1, T2 is 2, T3 is 16, and T4 is 272. It is read from the Bernoulli number B(2n). n runs fr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bernoulli-sum","url":"https://scorptech.it.com/api/x402/v1/bernoulli-sum","summary":"The sum of Bernoulli numbers B0 through Bn, as a reduced fraction. The sum through B1 is 3/2, and through B2 is 5/3. n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/isqrt","url":"https://scorptech.it.com/api/x402/v1/isqrt","summary":"The integer square root of a non-negative decimal integer, the greatest r with r^2 not exceeding n. isqrt(0) is 0 and is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/icbrt","url":"https://scorptech.it.com/api/x402/v1/icbrt","summary":"The integer cube root of a non-negative decimal integer, the greatest r with r^3 not exceeding n. icbrt(26) is 2 and icb","price":"$0.001","params":[]},{"method":"POST","path":"/v1/isquare","url":"https://scorptech.it.com/api/x402/v1/isquare","summary":"Whether a non-negative decimal integer is a perfect square, and the root when it is. 16 is 4 squared. 15 is not a square","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ipow","url":"https://scorptech.it.com/api/x402/v1/ipow","summary":"The integer power base^exp. 2^10 is 1024, and (-2)^5 is -32. 0^0 is 1. exp runs from 0 to 256. The base has at most 80 d","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perfect-power","url":"https://scorptech.it.com/api/x402/v1/perfect-power","summary":"Whether n is a perfect power a^k with k at least 2. The base is the smallest such a, so the exponent is the largest. 16 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/recaman","url":"https://scorptech.it.com/api/x402/v1/recaman","summary":"The nth Recamán term. a(0) is 0. Each later term steps back by n when that value is positive and new, and otherwise step","price":"$0.001","params":[]},{"method":"POST","path":"/v1/recaman-seen","url":"https://scorptech.it.com/api/x402/v1/recaman-seen","summary":"Whether value appears in Recamán's sequence from a(0) through a(limit), and the smallest index if it does. 2 appears at ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/golomb","url":"https://scorptech.it.com/api/x402/v1/golomb","summary":"The nth Golomb term, 1-based. a(1) is 1, a(n) counts how often n will appear, and a(6) is 4. The sequence starts 1, 2, 2","price":"$0.001","params":[]},{"method":"POST","path":"/v1/golomb-sum","url":"https://scorptech.it.com/api/x402/v1/golomb-sum","summary":"The sum of the first n Golomb terms, from a(1) through a(n). The sum of the first 5 terms is 11. n runs from 1 to 100000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/golomb-inverse","url":"https://scorptech.it.com/api/x402/v1/golomb-inverse","summary":"The smallest index n with Golomb a(n) equal to k. Index 1 holds 1, index 4 holds 3, and index 6 holds 4. k runs from 1 t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hockey-stick","url":"https://scorptech.it.com/api/x402/v1/hockey-stick","summary":"The hockey-stick sum of binomial coefficients C(i, r) from i = r through n. It equals C(n+1, r+1). The sum for r = 2 and","price":"$0.001","params":[]},{"method":"POST","path":"/v1/vandermonde","url":"https://scorptech.it.com/api/x402/v1/vandermonde","summary":"The Vandermonde convolution. The sum of C(m, k)*C(n, r-k) equals C(m+n, r). For m = 3, n = 3, and r = 2 the value is 15.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/stars-bars","url":"https://scorptech.it.com/api/x402/v1/stars-bars","summary":"The number of ways to place n identical items into k distinct bins, which is C(n+k-1, n). Three items in two bins can be","price":"$0.001","params":[]},{"method":"POST","path":"/v1/multichoose","url":"https://scorptech.it.com/api/x402/v1/multichoose","summary":"The number of ways to choose k elements from n types with repetition, which is C(n+k-1, k). Choosing 2 from 3 types can ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kummer","url":"https://scorptech.it.com/api/x402/v1/kummer","summary":"The number of carries when a and b are added in prime base p. By Kummer, that is the exponent of p in C(a+b, a). Adding ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/double-factorial","url":"https://scorptech.it.com/api/x402/v1/double-factorial","summary":"The double factorial n!!. Even n uses the even factors down to 2, and odd n uses the odd factors down to 1. 0!! is 1, 6!","price":"$0.001","params":[]},{"method":"POST","path":"/v1/primorial","url":"https://scorptech.it.com/api/x402/v1/primorial","summary":"The primorial of n, the product of the primes less than or equal to n. The primorial of 1 is 1, of 10 is 210, and of 11 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/falling","url":"https://scorptech.it.com/api/x402/v1/falling","summary":"The falling factorial (n)_k = n*(n-1)*...*(n-k+1). (5)_3 is 60, (5)_0 is 1, and (5)_6 is 0. n runs from 0 to 100000 and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/rising","url":"https://scorptech.it.com/api/x402/v1/rising","summary":"The rising factorial n^(k) = n*(n+1)*...*(n+k-1). 5^(3) is 210, and 5^(0) is 1. n runs from 0 to 100000 and k runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factorial-mod","url":"https://scorptech.it.com/api/x402/v1/factorial-mod","summary":"n! mod m. 0! is 1, so 0! mod 5 is 1. 10! mod 97 is 30. The modulus is a positive decimal integer, and zero is rejected. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sl2-mul","url":"https://scorptech.it.com/api/x402/v1/sl2-mul","summary":"The product of two integer 2 by 2 matrices, each written as [a, b, c, d]. [1, 1, 0, 1] times [1, 0, 1, 1] is [2, 1, 1, 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sl2-pow","url":"https://scorptech.it.com/api/x402/v1/sl2-pow","summary":"The non-negative integer power of a 2 by 2 integer matrix. The exponent 0 is the identity [1, 0, 0, 1]. [1, 1, 0, 1] to ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sl2-det","url":"https://scorptech.it.com/api/x402/v1/sl2-det","summary":"The determinant ad-bc of a 2 by 2 integer matrix [a, b, c, d]. The determinant of [1, 1, 0, 1] is 1, and the determinant","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sl2-inv","url":"https://scorptech.it.com/api/x402/v1/sl2-inv","summary":"The inverse of a 2 by 2 integer matrix whose determinant is 1 or -1. The inverse of [1, 1, 0, 1] is [1, -1, 0, 1]. Any o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sl2-trace","url":"https://scorptech.it.com/api/x402/v1/sl2-trace","summary":"The trace a+d of a 2 by 2 integer matrix [a, b, c, d]. The trace of [1, 1, 0, 1] is 2, and the trace of [8, 5, 5, 3] is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/frobenius","url":"https://scorptech.it.com/api/x402/v1/frobenius","summary":"The Frobenius number of two coprime integers a and b, each at least 2. It is ab-a-b, the largest integer that cannot be ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/coin-count","url":"https://scorptech.it.com/api/x402/v1/coin-count","summary":"How many positive integers cannot be made with coprime denominations a and b. The count is (a-1)*(b-1)/2. For 2 and 5 it","price":"$0.001","params":[]},{"method":"POST","path":"/v1/coin-check","url":"https://scorptech.it.com/api/x402/v1/coin-check","summary":"Whether n can be made with non-negative counts of coins a and b. 10 can be made with 2 and 5. 3 cannot. 0 can. a and b r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/coin-ways","url":"https://scorptech.it.com/api/x402/v1/coin-ways","summary":"The number of non-negative integer solutions of a*x + b*y = n. Order of summands does not matter. There are 2 ways to ma","price":"$0.001","params":[]},{"method":"POST","path":"/v1/coin-min","url":"https://scorptech.it.com/api/x402/v1/coin-min","summary":"The fewest coins of denominations a and b that sum to n. Making 10 with 2 and 5 takes 2 coins. An amount that cannot be ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/look-say","url":"https://scorptech.it.com/api/x402/v1/look-say","summary":"The nth look-and-say term, starting from 1. Term 1 is 1, term 4 is 1211, and term 5 is 111221. Each run of k copies of a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/look-say-step","url":"https://scorptech.it.com/api/x402/v1/look-say-step","summary":"One look-and-say step on a digit string. 1 becomes 11, and 1211 becomes 111221. Two 2s become 22. An empty string, or a ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/look-say-len","url":"https://scorptech.it.com/api/x402/v1/look-say-len","summary":"The number of digits in the nth look-and-say term, starting from 1. Term 1 has length 1, term 5 has length 6, and term 8","price":"$0.001","params":[]},{"method":"POST","path":"/v1/look-say-digit","url":"https://scorptech.it.com/api/x402/v1/look-say-digit","summary":"One digit of the nth look-and-say term. The index starts at 0. Term 5 is 111221, so index 3 is 2. An index past the end ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/look-say-count","url":"https://scorptech.it.com/api/x402/v1/look-say-count","summary":"How many times a digit occurs in the nth look-and-say term. Term 5 is 111221, so it has four 1s and two 2s. A digit from","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-q","url":"https://scorptech.it.com/api/x402/v1/hof-q","summary":"The nth Hofstadter Q term. Q(1) and Q(2) are 1, and Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2)). Q(10) is 6, and Q(16) is 9. n runs","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-q-sum","url":"https://scorptech.it.com/api/x402/v1/hof-q-sum","summary":"The sum of Hofstadter Q terms Q(1) through Q(n). The sum through Q(10) is 36, and the sum through Q(20) is 129. n runs f","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-q-prefix","url":"https://scorptech.it.com/api/x402/v1/hof-q-prefix","summary":"The first n Hofstadter Q terms, from Q(1) through Q(n), written as comma-separated integers. The first 10 terms are 1,1,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-g","url":"https://scorptech.it.com/api/x402/v1/hof-g","summary":"The nth Hofstadter G term. G(0) is 0, and G(n) = n - G(G(n-1)). G(1) is 1, and G(10) is 6. n runs from 0 to 100000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-h","url":"https://scorptech.it.com/api/x402/v1/hof-h","summary":"The nth Hofstadter H term. H(0) is 0, and H(n) = n - H(H(H(n-1))). H(1) is 1, and H(10) is 7. n runs from 0 to 100000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-conway","url":"https://scorptech.it.com/api/x402/v1/hof-conway","summary":"The nth Hofstadter-Conway term. a(1) and a(2) are 1, and a(n) = a(a(n-1)) + a(n-a(n-1)). a(5) is 3, and a(10) is 6. n ru","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-conway-sum","url":"https://scorptech.it.com/api/x402/v1/hof-conway-sum","summary":"The sum of Hofstadter-Conway terms a(1) through a(n). The sum through a(10) is 32, and the sum through a(30) is 266. n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-male","url":"https://scorptech.it.com/api/x402/v1/hof-male","summary":"The nth Hofstadter male term. M(0) is 0, F(0) is 1, M(n) = n - F(M(n-1)), and F(n) = n - M(F(n-1)). M(2) is 1, and M(5) ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-female","url":"https://scorptech.it.com/api/x402/v1/hof-female","summary":"The nth Hofstadter female term. F(0) is 1, M(0) is 0, F(n) = n - M(F(n-1)), and M(n) = n - F(M(n-1)). F(1) is 1, and F(5","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hof-pair","url":"https://scorptech.it.com/api/x402/v1/hof-pair","summary":"The Hofstadter male and female terms at the same index. M(0) is 0 and F(0) is 1. At n = 5 both terms are 3. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/tetranacci","url":"https://scorptech.it.com/api/x402/v1/tetranacci","summary":"The nth Tetranacci number. T0, T1, and T2 are 0, T3 is 1, and each later term is the sum of the previous four. T(10) is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/tetranacci-sum","url":"https://scorptech.it.com/api/x402/v1/tetranacci-sum","summary":"The sum of Tetranacci numbers from lo through hi inclusive. The sum from T0 through T10 is 116. lo must not pass hi. Eac","price":"$0.001","params":[]},{"method":"POST","path":"/v1/tetranacci-mod","url":"https://scorptech.it.com/api/x402/v1/tetranacci-mod","summary":"T(n) mod m for the Tetranacci sequence. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/tetranacci-index","url":"https://scorptech.it.com/api/x402/v1/tetranacci-index","summary":"The smallest index n with T(n) equal to value. 0 is index 0, and 1 is index 3. A number that is not a Tetranacci term, o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/tetranacci-step","url":"https://scorptech.it.com/api/x402/v1/tetranacci-step","summary":"The Tetranacci step T(n) = T(n-1) + T(n-2) + T(n-3) + T(n-4). a is T(n-4), b is T(n-3), c is T(n-2), and d is T(n-1). n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pentanacci","url":"https://scorptech.it.com/api/x402/v1/pentanacci","summary":"The nth Pentanacci number. P0 through P3 are 0, P4 is 1, and each later term is the sum of the previous five. P(10) is 3","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pentanacci-sum","url":"https://scorptech.it.com/api/x402/v1/pentanacci-sum","summary":"The sum of Pentanacci numbers from lo through hi inclusive. The sum from P0 through P12 is 244. lo must not pass hi. Eac","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pentanacci-mod","url":"https://scorptech.it.com/api/x402/v1/pentanacci-mod","summary":"P(n) mod m for the Pentanacci sequence. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pentanacci-index","url":"https://scorptech.it.com/api/x402/v1/pentanacci-index","summary":"The smallest index n with P(n) equal to value. 0 is index 0, and 1 is index 4. A number that is not a Pentanacci term, o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pentanacci-step","url":"https://scorptech.it.com/api/x402/v1/pentanacci-step","summary":"The Pentanacci step P(n) = P(n-1) + P(n-2) + P(n-3) + P(n-4) + P(n-5). a is P(n-5), b is P(n-4), c is P(n-3), d is P(n-2","price":"$0.001","params":[]},{"method":"POST","path":"/v1/leonardo","url":"https://scorptech.it.com/api/x402/v1/leonardo","summary":"The nth Leonardo number. L(0) and L(1) are 1, and L(n) = L(n-1) + L(n-2) + 1. L(4) is 9, and L(10) is 177. An optional m","price":"$0.001","params":[]},{"method":"POST","path":"/v1/leonardo-sum","url":"https://scorptech.it.com/api/x402/v1/leonardo-sum","summary":"The sum of Leonardo numbers from lo through hi inclusive. The sum from L(0) through L(5) is 34. lo must not pass hi. Eac","price":"$0.001","params":[]},{"method":"POST","path":"/v1/leonardo-mod","url":"https://scorptech.it.com/api/x402/v1/leonardo-mod","summary":"L(n) mod m for the Leonardo sequence. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 to ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/leonardo-index","url":"https://scorptech.it.com/api/x402/v1/leonardo-index","summary":"The smallest index n with L(n) equal to value. 1 is index 0, and 9 is index 4. A number that is not a Leonardo number, o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/leonardo-fib","url":"https://scorptech.it.com/api/x402/v1/leonardo-fib","summary":"The Leonardo identity L(n) = 2*F(n+1) - 1. F is the Fibonacci sequence with F(0) = 0 and F(1) = 1. For n = 10, L is 177 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/euler-zigzag","url":"https://scorptech.it.com/api/x402/v1/euler-zigzag","summary":"The nth Euler zigzag number, the number of up-down permutations of n. a(0) is 1, a(3) is 2, a(4) is 5, and a(5) is 16. F","price":"$0.001","params":[]},{"method":"POST","path":"/v1/euler-zigzag-sum","url":"https://scorptech.it.com/api/x402/v1/euler-zigzag-sum","summary":"The sum of Euler zigzag numbers a(0) through a(n). The sum through a(5) is 26, and the sum through a(10) is 60201. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/entringer","url":"https://scorptech.it.com/api/x402/v1/entringer","summary":"The Entringer number E(n, k). E(1, 1) is 1, and E(n, k) = E(n, k-1) + E(n-1, n-k). E(5, 2) is 10, and E(6, 3) is 46. The","price":"$0.001","params":[]},{"method":"POST","path":"/v1/genocchi","url":"https://scorptech.it.com/api/x402/v1/genocchi","summary":"The nth Genocchi number |G(2n)|, from G(2n) = 2*(1-4^n)*B(2n) with B(1) = +1/2. The first four absolute values are 1, 1,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/euler-secant","url":"https://scorptech.it.com/api/x402/v1/euler-secant","summary":"The nth secant number, the signed Euler number E(2n) = (-1)^n * a(2n). E(0) is 1, E(2) is -1, E(4) is 5, and E(6) is -61","price":"$0.001","params":[]},{"method":"POST","path":"/v1/involution","url":"https://scorptech.it.com/api/x402/v1/involution","summary":"The number of involutions of n objects. I(0) and I(1) are 1, and I(n) = I(n-1) + (n-1)*I(n-2). I(5) is 26, and I(10) is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/involution-sum","url":"https://scorptech.it.com/api/x402/v1/involution-sum","summary":"The sum of involution numbers from lo through hi inclusive. The sum from I(0) through I(10) is 13232. lo must not pass h","price":"$0.001","params":[]},{"method":"POST","path":"/v1/involution-mod","url":"https://scorptech.it.com/api/x402/v1/involution-mod","summary":"I(n) mod m, the involution number reduced by a positive modulus. Zero is rejected. n runs from 0 to 100. The residue is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/rencontres","url":"https://scorptech.it.com/api/x402/v1/rencontres","summary":"The rencontres number D(n, k), the permutations of n objects with k fixed points. D(n, k) = C(n, k) * !(n-k). D(5, 0) is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/involution-fixed","url":"https://scorptech.it.com/api/x402/v1/involution-fixed","summary":"The number of involutions of n objects with exactly k fixed points. It is 0 when n-k is odd. For n = 5 and k = 1 the cou","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fubini","url":"https://scorptech.it.com/api/x402/v1/fubini","summary":"The nth Fubini number, the ordered partitions of an n-element set. a(0) is 1, a(2) is 3, a(3) is 13, and a(5) is 541. n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fubini-sum","url":"https://scorptech.it.com/api/x402/v1/fubini-sum","summary":"The sum of Fubini numbers from lo through hi inclusive. The sum from a(0) through a(5) is 634. lo must not pass hi. Each","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fubini-mod","url":"https://scorptech.it.com/api/x402/v1/fubini-mod","summary":"The nth Fubini number mod m. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 to 20. The r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lah","url":"https://scorptech.it.com/api/x402/v1/lah","summary":"The unsigned Lah number L(n, k). L(0, 0) is 1, L(n, 0) is 0 for n > 0, and L(n, 1) is n!. L(5, 2) is 240, and L(6, 3) is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lah-row","url":"https://scorptech.it.com/api/x402/v1/lah-row","summary":"Row n of the unsigned Lah numbers, from L(n, 0) through L(n, n), written as comma-separated integers. Row 5 is 0,120,240","price":"$0.001","params":[]},{"method":"POST","path":"/v1/riordan","url":"https://scorptech.it.com/api/x402/v1/riordan","summary":"The nth Riordan number. R(0) is 1, R(1) is 0, and (n+1)*R(n) = (n-1)*(2*R(n-1) + 3*R(n-2)) for n at least 2. R(4) is 3, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/riordan-sum","url":"https://scorptech.it.com/api/x402/v1/riordan-sum","summary":"The sum of Riordan numbers from lo through hi inclusive. The sum from R(0) through R(10) is 989. lo must not pass hi. Ea","price":"$0.001","params":[]},{"method":"POST","path":"/v1/riordan-mod","url":"https://scorptech.it.com/api/x402/v1/riordan-mod","summary":"R(n) mod m for the Riordan numbers. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 to 80","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fine","url":"https://scorptech.it.com/api/x402/v1/fine","summary":"The nth Fine number. F(0) is 1, F(1) is 0, F(2) is 1, and F(6) is 57. For n at least 1, F(n) = C(n) - sum of C(i)*F(n-1-","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fine-sum","url":"https://scorptech.it.com/api/x402/v1/fine-sum","summary":"The sum of Fine numbers from lo through hi inclusive. The sum from F(0) through F(6) is 85. lo must not pass hi. Each in","price":"$0.001","params":[]},{"method":"POST","path":"/v1/period-double","url":"https://scorptech.it.com/api/x402/v1/period-double","summary":"The period-doubling bit at index n, starting at 0. It is the parity of the exponent of 2 in n+1. Bit 0 is 0, bit 1 is 1,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/period-double-prefix","url":"https://scorptech.it.com/api/x402/v1/period-double-prefix","summary":"The first n period-doubling bits, as a string of the digits 0 and 1. The first 8 bits are 01000101. n runs from 1 to 409","price":"$0.001","params":[]},{"method":"POST","path":"/v1/period-double-sum","url":"https://scorptech.it.com/api/x402/v1/period-double-sum","summary":"The number of 1s in the first n period-doubling bits, from index 0 through n-1. The first 8 bits have three 1s, and the ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cantor-bit","url":"https://scorptech.it.com/api/x402/v1/cantor-bit","summary":"The Cantor bit of n. It is 1 when every base-3 digit of n is 0 or 1, and 0 when a digit is 2. Bit 0 is 1, bit 2 is 0, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cantor-count","url":"https://scorptech.it.com/api/x402/v1/cantor-count","summary":"How many integers from 0 through n have only the digits 0 and 1 in base 3. The count through 0 is 1, through 2 is 2, and","price":"$0.001","params":[]},{"method":"POST","path":"/v1/two-squares","url":"https://scorptech.it.com/api/x402/v1/two-squares","summary":"How many integer pairs (x, y), counting order and signs, satisfy x^2 + y^2 = n. There is 1 for 0, 4 for 1, and 8 for 5. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/two-squares-pair","url":"https://scorptech.it.com/api/x402/v1/two-squares-pair","summary":"Nonnegative integers x and y with x <= y and x^2 + y^2 = n, choosing the greatest y. 5 is 1 and 2. 25 is 0 and 5. A numb","price":"$0.001","params":[]},{"method":"POST","path":"/v1/three-squares","url":"https://scorptech.it.com/api/x402/v1/three-squares","summary":"Whether n is a sum of three integer squares, and one nonnegative witness x <= y <= z when it is. It is impossible exactl","price":"$0.001","params":[]},{"method":"POST","path":"/v1/four-squares","url":"https://scorptech.it.com/api/x402/v1/four-squares","summary":"Four nonnegative integers w <= x <= y <= z with w^2 + x^2 + y^2 + z^2 = n. Every nonnegative integer has one. 7 is 1, 1,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/square-obstruction","url":"https://scorptech.it.com/api/x402/v1/square-obstruction","summary":"Write n = 4^power * odd, with odd not divisible by 4. blocked is true when odd is congruent to 7 mod 8, which is exactly","price":"$0.001","params":[]},{"method":"POST","path":"/v1/liouville","url":"https://scorptech.it.com/api/x402/v1/liouville","summary":"Liouville's function lambda(n) = (-1) to the number of prime factors of n, counted with multiplicity. lambda(1) is 1, la","price":"$0.001","params":[]},{"method":"POST","path":"/v1/liouville-sum","url":"https://scorptech.it.com/api/x402/v1/liouville-sum","summary":"The sum of Liouville's lambda(k) from 1 through n. The sum through 10 is 0, and the sum through 30 is -4. n runs from 1 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mertens","url":"https://scorptech.it.com/api/x402/v1/mertens","summary":"The Mertens function M(n), the sum of the Mobius function mu(k) from 1 through n. M(1) is 1, M(10) is -1, and M(100) is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mertens-max","url":"https://scorptech.it.com/api/x402/v1/mertens-max","summary":"The largest absolute value of the Mertens function M(k) for k from 1 through n, and the smallest k where it occurs. Thro","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prime-power","url":"https://scorptech.it.com/api/x402/v1/prime-power","summary":"If n is a prime power p^k with k at least 1, return that prime and the exponent. Otherwise the prime and the exponent ar","price":"$0.001","params":[]},{"method":"POST","path":"/v1/square-part","url":"https://scorptech.it.com/api/x402/v1/square-part","summary":"The largest square dividing n. 1 gives 1, 8 gives 4, 12 gives 4, and 18 gives 9. n runs from 1 to 1000000000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/square-free-part","url":"https://scorptech.it.com/api/x402/v1/square-free-part","summary":"n divided by its largest square factor. 12 gives 3, 18 gives 2, 36 gives 1, and 45 gives 5. n runs from 1 to 10000000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/powerful","url":"https://scorptech.it.com/api/x402/v1/powerful","summary":"Whether n is powerful: 1, or every prime factor's exponent is at least 2. 1, 8, and 36 are powerful. 12 is not. n runs f","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cubefull","url":"https://scorptech.it.com/api/x402/v1/cubefull","summary":"Whether n is cubefull: 1, or every prime factor's exponent is at least 3. 1 and 8 are cubefull. 36 is powerful but not c","price":"$0.001","params":[]},{"method":"POST","path":"/v1/achilles","url":"https://scorptech.it.com/api/x402/v1/achilles","summary":"Whether n is an Achilles number: powerful, but not a perfect power. 72 and 108 are Achilles. 36 is a square, and 1 is a ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sigma-k","url":"https://scorptech.it.com/api/x402/v1/sigma-k","summary":"The sum of the kth powers of the divisors of n. sigma_2(1) is 1, sigma_2(6) is 50, and sigma_2(12) is 210. n runs from 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/abundancy","url":"https://scorptech.it.com/api/x402/v1/abundancy","summary":"The abundancy sigma(n)/n as a reduced fraction. 1 is 1/1, 4 is 7/4, 6 is 2/1, and 12 is 7/3. n runs from 1 to 1000000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/deficiency","url":"https://scorptech.it.com/api/x402/v1/deficiency","summary":"The deficiency 2n - sigma(n). A deficient number is positive, a perfect number is 0, and an abundant number is negative.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/unitary-sigma","url":"https://scorptech.it.com/api/x402/v1/unitary-sigma","summary":"The sum of the unitary divisors of n. A unitary divisor shares no prime factor with the complementary divisor. The sum i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/unitary-count","url":"https://scorptech.it.com/api/x402/v1/unitary-count","summary":"How many unitary divisors n has. The count is 2 to the number of distinct prime factors, and 1 has one unitary divisor. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/happy","url":"https://scorptech.it.com/api/x402/v1/happy","summary":"Follow the happy-number map: replace n by the sum of the squares of its decimal digits. 1 is already happy. 7 reaches 1 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/harshad","url":"https://scorptech.it.com/api/x402/v1/harshad","summary":"Whether n is a base-10 Harshad number: n is divisible by the sum of its decimal digits. 18 is Harshad, and 11 is not. n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/harshad-base","url":"https://scorptech.it.com/api/x402/v1/harshad-base","summary":"Whether n is a Harshad number in base b: n is divisible by the sum of its base-b digits. 10 is Harshad in base 2. 13 is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/additive-persistence","url":"https://scorptech.it.com/api/x402/v1/additive-persistence","summary":"Additive persistence: how many times the decimal digits of n must be summed to reach a single digit, and that digit. 199","price":"$0.001","params":[]},{"method":"POST","path":"/v1/multiplicative-persistence","url":"https://scorptech.it.com/api/x402/v1/multiplicative-persistence","summary":"Multiplicative persistence: how many times the decimal digits of n must be multiplied to reach a single digit, and that ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wedderburn","url":"https://scorptech.it.com/api/x402/v1/wedderburn","summary":"The nth Wedderburn-Etherington number. a(1) is 1, and a(1) through a(6) are 1, 1, 1, 2, 3, 6. Even and odd indices use t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wedderburn-sum","url":"https://scorptech.it.com/api/x402/v1/wedderburn-sum","summary":"The sum of Wedderburn-Etherington numbers from lo through hi inclusive. The sum from a(1) through a(6) is 14. lo must no","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wedderburn-mod","url":"https://scorptech.it.com/api/x402/v1/wedderburn-mod","summary":"a(n) mod m for the Wedderburn-Etherington numbers. The modulus is a positive decimal integer, and zero is rejected. n ru","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wedderburn-index","url":"https://scorptech.it.com/api/x402/v1/wedderburn-index","summary":"The smallest index n with Wedderburn-Etherington number a(n) equal to value. 1 is index 1, and 6 is index 6. A number th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wedderburn-split","url":"https://scorptech.it.com/api/x402/v1/wedderburn-split","summary":"The Wedderburn-Etherington split of n. cross is the sum of a(i)*a(n-i) over the pairs below the middle, and middle is a(","price":"$0.001","params":[]},{"method":"POST","path":"/v1/alt-factorial","url":"https://scorptech.it.com/api/x402/v1/alt-factorial","summary":"The alternating factorial af(n) = n! - (n-1)! + (n-2)! - ... ending at 1. af(1) is 1, af(3) is 5, and af(4) is 19. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/alt-sum","url":"https://scorptech.it.com/api/x402/v1/alt-sum","summary":"The sum of alternating factorials from lo through hi inclusive. The sum from af(1) through af(5) is 127. lo must not pas","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kurepa","url":"https://scorptech.it.com/api/x402/v1/kurepa","summary":"The Kurepa function K(n), the sum of k! for k from 0 through n-1. K(1) is 1, K(2) is 2, and K(5) is 34. This is the left","price":"$0.001","params":[]},{"method":"POST","path":"/v1/superfactorial","url":"https://scorptech.it.com/api/x402/v1/superfactorial","summary":"The superfactorial sf(n), the product of k! for k from 1 through n. sf(1) is 1, sf(3) is 12, and sf(4) is 288. n runs fr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kurepa-mod","url":"https://scorptech.it.com/api/x402/v1/kurepa-mod","summary":"K(n) mod m for the Kurepa function, the sum of k! from 0 through n-1. The modulus is a positive decimal integer, and zer","price":"$0.001","params":[]},{"method":"POST","path":"/v1/next-prime","url":"https://scorptech.it.com/api/x402/v1/next-prime","summary":"The least prime greater than or equal to n. 0, 1, and 2 all give 2. 14 gives 17. 10000000 gives 10000019. n runs from 0 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prev-prime","url":"https://scorptech.it.com/api/x402/v1/prev-prime","summary":"The greatest prime less than or equal to n. 2 gives 2, 10 gives 7, and 10000000 gives 9999991. n runs from 2 to 10000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prime-count","url":"https://scorptech.it.com/api/x402/v1/prime-count","summary":"pi(n), the number of primes less than or equal to n. pi(0) and pi(1) are 0, pi(10) is 4, pi(100) is 25, and pi(10000000)","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prime-nth","url":"https://scorptech.it.com/api/x402/v1/prime-nth","summary":"The kth prime. The 1st is 2, the 10th is 29, the 100th is 541, and the 664579th is 9999991. k runs from 1 to 664579.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prime-gap","url":"https://scorptech.it.com/api/x402/v1/prime-gap","summary":"The prime at or below n, the next prime after that one, and the difference. At 10 the prime is 7, the next is 11, and th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/popcount","url":"https://scorptech.it.com/api/x402/v1/popcount","summary":"The number of 1 bits in n. 0 has none, 255 has 8, and 256 has 1. n runs from 0 to 999999999999999.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bit-length","url":"https://scorptech.it.com/api/x402/v1/bit-length","summary":"The bit length of n: 0 for 0, and one more than the position of the highest 1 bit otherwise. 1 has length 1, 255 has len","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bit-reverse","url":"https://scorptech.it.com/api/x402/v1/bit-reverse","summary":"Reverse the lowest width bits of n. 13 reversed in width 4 is 11. 1 reversed in width 8 is 128. n must be smaller than 2","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hamming-dist","url":"https://scorptech.it.com/api/x402/v1/hamming-dist","summary":"The Hamming distance of a and b: how many bit positions differ. 13 and 7 differ in 2 places. 0 and 0 differ in none. Bot","price":"$0.001","params":[]},{"method":"POST","path":"/v1/bit-subset","url":"https://scorptech.it.com/api/x402/v1/bit-subset","summary":"Whether every 1 bit of a is also a 1 bit of b. 3 is a bit-subset of 7. 3 is not a bit-subset of 5. 0 is a bit-subset of ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kronecker","url":"https://scorptech.it.com/api/x402/v1/kronecker","summary":"The Kronecker symbol (a/n) for any integers a and n. (2/7) is 1, (3/7) is -1, (2/4) is 0, and (1/0) is 1. Both a and n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/legendre-symbol","url":"https://scorptech.it.com/api/x402/v1/legendre-symbol","summary":"The Legendre symbol (a/p) for an odd prime p. (2/7) is 1, (3/7) is -1, and (-1/5) is 1. p must be an odd prime. a runs f","price":"$0.001","params":[]},{"method":"POST","path":"/v1/quadratic-residue","url":"https://scorptech.it.com/api/x402/v1/quadratic-residue","summary":"Whether a is a square modulo an odd prime p, counting 0 as a square. 2 is a square mod 7, and 3 is not. -1 is a square m","price":"$0.001","params":[]},{"method":"POST","path":"/v1/padic","url":"https://scorptech.it.com/api/x402/v1/padic","summary":"The p-adic valuation v_p(n): how many times the prime p divides n. v_2(12) is 2, v_3(12) is 1, and v_5(12) is 0. p must ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factorial-valuation","url":"https://scorptech.it.com/api/x402/v1/factorial-valuation","summary":"The p-adic valuation of n!, the exponent of the prime p in n!. v_2(10!) is 8, v_3(10!) is 4, and v_2(100!) is 97. p must","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mult-order","url":"https://scorptech.it.com/api/x402/v1/mult-order","summary":"The multiplicative order of a modulo n: the least positive k with a^k = 1 mod n. The order of 2 mod 7 is 3, and the orde","price":"$0.001","params":[]},{"method":"POST","path":"/v1/primitive-root","url":"https://scorptech.it.com/api/x402/v1/primitive-root","summary":"Whether g is a primitive root modulo the prime p. 3 is a primitive root mod 7, and 2 is not. 1 is a primitive root mod 2","price":"$0.001","params":[]},{"method":"POST","path":"/v1/least-primitive-root","url":"https://scorptech.it.com/api/x402/v1/least-primitive-root","summary":"The least primitive root modulo the prime p. It is 1 for 2, 3 for 7, 5 for 23, and 6 for 41. p must be prime and runs fr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mod-inverse","url":"https://scorptech.it.com/api/x402/v1/mod-inverse","summary":"The inverse of a modulo m, the residue from 0 to m-1 with a*inverse = 1 mod m. The inverse of 3 mod 7 is 5, and the inve","price":"$0.001","params":[]},{"method":"POST","path":"/v1/power-mod","url":"https://scorptech.it.com/api/x402/v1/power-mod","summary":"a to the power e, reduced modulo m. 2^10 mod 1000 is 24, 3^5 mod 7 is 5, and (-2)^3 mod 7 is 6. A zero exponent gives 1,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/eulerian","url":"https://scorptech.it.com/api/x402/v1/eulerian","summary":"The Eulerian number A(n, k), the number of permutations of n with k ascents. A(4, 1) is 11, A(5, 2) is 66, and A(n, 0) i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/eulerian-row","url":"https://scorptech.it.com/api/x402/v1/eulerian-row","summary":"The Eulerian row A(n, 0) through A(n, n-1), comma-separated. The row for 5 is 1,26,66,26,1. It is symmetric and sums to ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/eulerian-sum","url":"https://scorptech.it.com/api/x402/v1/eulerian-sum","summary":"The sum of Eulerian numbers A(n, k) for k from lo through hi. The whole row sums to n!. A(4, 0) through A(4, 1) is 12. l","price":"$0.001","params":[]},{"method":"POST","path":"/v1/eulerian-mod","url":"https://scorptech.it.com/api/x402/v1/eulerian-mod","summary":"A(n, k) mod m for the Eulerian numbers. The modulus is a positive decimal integer, and zero is rejected. k must be an as","price":"$0.001","params":[]},{"method":"POST","path":"/v1/eulerian-poly","url":"https://scorptech.it.com/api/x402/v1/eulerian-poly","summary":"The Eulerian polynomial A_n(x) = sum A(n, k) * x^k. A_3(2) is 13, A_4(1) is 24, and A_5(0) is 1. n runs from 1 to 20, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/central-binomial","url":"https://scorptech.it.com/api/x402/v1/central-binomial","summary":"The nth central binomial coefficient (2n)!/(n!)^2. C(0) is 1, C(1) is 2, C(5) is 252, and C(10) is 184756. n runs from 0","price":"$0.001","params":[]},{"method":"POST","path":"/v1/central-sum","url":"https://scorptech.it.com/api/x402/v1/central-sum","summary":"The sum of central binomial coefficients from lo through hi. The sum from C(0) through C(5) is 351. lo must not pass hi.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/central-mod","url":"https://scorptech.it.com/api/x402/v1/central-mod","summary":"C(n) mod m for the central binomial coefficients. C(10) mod 7 is 5. The modulus is a positive decimal integer, and zero ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/central-index","url":"https://scorptech.it.com/api/x402/v1/central-index","summary":"The index n with central binomial coefficient C(n) equal to value. 1 is index 0, and 2 is index 1. A number that is not ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/central-ratio","url":"https://scorptech.it.com/api/x402/v1/central-ratio","summary":"The ratio C(n)/C(n-1) = (4n-2)/n in lowest terms. For 1 it is 2/1, for 2 it is 3/1, and for 3 it is 10/3. n runs from 1 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/practical","url":"https://scorptech.it.com/api/x402/v1/practical","summary":"Whether n is practical: every integer from 1 to n is a sum of distinct divisors of n. 1, 2, 6, and 12 are practical. 3 a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/practical-count","url":"https://scorptech.it.com/api/x402/v1/practical-count","summary":"How many practical numbers are from 1 through n. There are 5 through 10, and 15 through 40. n runs from 1 to 5000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/semiperfect","url":"https://scorptech.it.com/api/x402/v1/semiperfect","summary":"Whether some subset of the proper divisors of n sums to n. 6 and 12 are semiperfect. 10 and 70 are not. n runs from 1 to","price":"$0.001","params":[]},{"method":"POST","path":"/v1/weird","url":"https://scorptech.it.com/api/x402/v1/weird","summary":"Whether n is weird: abundant, but not semiperfect. 70 is weird, and so is 836. 12 is abundant but semiperfect, so it is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/amicable","url":"https://scorptech.it.com/api/x402/v1/amicable","summary":"Whether n is amicable, and its partner when it is. The partner of 220 is 284, and the partner of 284 is 220. A perfect n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/landau","url":"https://scorptech.it.com/api/x402/v1/landau","summary":"Landau's function g(n), the largest least common multiple of the parts of a partition of n. g(0) is 1, g(5) is 6, g(7) i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/landau-sum","url":"https://scorptech.it.com/api/x402/v1/landau-sum","summary":"The sum of Landau's function from lo through hi. The sum from g(0) through g(10) is 100. lo must not pass hi. Each index","price":"$0.001","params":[]},{"method":"POST","path":"/v1/landau-mod","url":"https://scorptech.it.com/api/x402/v1/landau-mod","summary":"g(n) mod m for Landau's function. g(12) mod 7 is 4, and g(25) mod 97 is 96. The modulus is a positive decimal integer, a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/landau-index","url":"https://scorptech.it.com/api/x402/v1/landau-index","summary":"The smallest n with Landau value g(n) equal to value. 1 is index 0, and 6 is index 5. A number that is not a value, or a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/range-lcm","url":"https://scorptech.it.com/api/x402/v1/range-lcm","summary":"The least common multiple of the integers from lo through hi. The LCM of 1 through 10 is 2520, and the LCM of 1 through ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucky","url":"https://scorptech.it.com/api/x402/v1/lucky","summary":"Whether n is a lucky number. The list begins 1, 3, 7, 9, 13, 15, 21. 5 and every even number are not lucky. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucky-nth","url":"https://scorptech.it.com/api/x402/v1/lucky-nth","summary":"The kth lucky number. The 1st is 1, the 10th is 33, and the 20th is 79. k runs from 1 to 8772, which is the count of luc","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucky-count","url":"https://scorptech.it.com/api/x402/v1/lucky-count","summary":"How many lucky numbers are from 1 through n. There is 1 through 1, 2 through 3, and 13 through 50. n runs from 1 to 1000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucky-next","url":"https://scorptech.it.com/api/x402/v1/lucky-next","summary":"The least lucky number greater than or equal to n. 2 gives 3, 20 gives 21, and 49 gives 49. n runs from 1 to 99987, the ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lucky-twin","url":"https://scorptech.it.com/api/x402/v1/lucky-twin","summary":"Whether n and n+2 are both lucky. 1, 7, and 13 start a lucky pair. 3 does not, because 5 is not lucky. n runs from 1 to ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hexanacci","url":"https://scorptech.it.com/api/x402/v1/hexanacci","summary":"The nth hexanacci number. H(0) through H(4) are 0, H(5) is 1, and H(10) is 16. Each later term adds the previous six. n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hexanacci-sum","url":"https://scorptech.it.com/api/x402/v1/hexanacci-sum","summary":"The sum of hexanacci numbers from lo through hi. The sum from H(0) through H(10) is 32. lo must not pass hi. Each index ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hexanacci-mod","url":"https://scorptech.it.com/api/x402/v1/hexanacci-mod","summary":"H(n) mod m for the hexanacci numbers. H(10) mod 7 is 2, and H(57) mod 97 is 61. The modulus is a positive decimal intege","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hexanacci-index","url":"https://scorptech.it.com/api/x402/v1/hexanacci-index","summary":"The smallest index n with hexanacci number H(n) equal to value. 0 is index 0, and 1 is index 5. A number that is not a t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hexanacci-step","url":"https://scorptech.it.com/api/x402/v1/hexanacci-step","summary":"The hexanacci step H(n) = H(n-1) + H(n-2) + H(n-3) + H(n-4) + H(n-5) + H(n-6). a through f are those six earlier terms, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/largest-prime-factor","url":"https://scorptech.it.com/api/x402/v1/largest-prime-factor","summary":"The largest prime factor of n. It is 1 for 1, 3 for 12, 5 for 30, and 7 for 35. n runs from 1 to 1000000000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/smallest-prime-factor","url":"https://scorptech.it.com/api/x402/v1/smallest-prime-factor","summary":"The smallest prime factor of n. It is 1 for 1, 2 for 12, 5 for 35, and 7 for 49. n runs from 1 to 1000000000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/smooth","url":"https://scorptech.it.com/api/x402/v1/smooth","summary":"Whether n is y-smooth: its largest prime factor is at most y. 12 is 3-smooth and not 2-smooth. 1 is 1-smooth. n and y ru","price":"$0.001","params":[]},{"method":"POST","path":"/v1/rough","url":"https://scorptech.it.com/api/x402/v1/rough","summary":"Whether n is y-rough: its smallest prime factor is at least y. 35 is 5-rough and not 7-rough. 1 is 1-rough and not 2-rou","price":"$0.001","params":[]},{"method":"POST","path":"/v1/powersmooth","url":"https://scorptech.it.com/api/x402/v1/powersmooth","summary":"Whether n is y-power-smooth: every maximal prime-power factor p^k is at most y. 12 is 4-power-smooth and not 3-power-smo","price":"$0.001","params":[]},{"method":"POST","path":"/v1/highly-composite","url":"https://scorptech.it.com/api/x402/v1/highly-composite","summary":"Whether n is highly composite: it has strictly more divisors than every smaller positive integer. 1, 2, 6, 12, 60, and 5","price":"$0.001","params":[]},{"method":"POST","path":"/v1/highly-composite-next","url":"https://scorptech.it.com/api/x402/v1/highly-composite-next","summary":"The least highly composite number greater than or equal to n. 1 gives 1, 3 gives 4, 7 gives 12, and 5041 gives 7560. n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/highly-composite-count","url":"https://scorptech.it.com/api/x402/v1/highly-composite-count","summary":"How many highly composite numbers are from 1 through n. There are 4 through 10, 9 through 60, and 10 through 120. n runs","price":"$0.001","params":[]},{"method":"POST","path":"/v1/divisor-record","url":"https://scorptech.it.com/api/x402/v1/divisor-record","summary":"The largest number of divisors attained by any integer from 1 through n, and the smallest integer that attains it. Throu","price":"$0.001","params":[]},{"method":"POST","path":"/v1/refactorable","url":"https://scorptech.it.com/api/x402/v1/refactorable","summary":"Whether n is refactorable: the number of positive divisors of n divides n. 1, 2, 8, 9, and 12 are refactorable. 4 and 6 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/subtraction-grundy","url":"https://scorptech.it.com/api/x402/v1/subtraction-grundy","summary":"The Grundy value of a heap of n in the subtraction game with the given move sizes. With moves 1, 2, and 3, the value of ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/subtraction-win","url":"https://scorptech.it.com/api/x402/v1/subtraction-win","summary":"Whether the player to move wins a subtraction game on a heap of n with perfect play. A win is a nonzero Grundy value. Wi","price":"$0.001","params":[]},{"method":"POST","path":"/v1/subtraction-row","url":"https://scorptech.it.com/api/x402/v1/subtraction-row","summary":"Grundy values from heap 0 through heap n in a subtraction game, comma-separated. With moves 1, 2, and 3, the row through","price":"$0.001","params":[]},{"method":"POST","path":"/v1/nim-sum","url":"https://scorptech.it.com/api/x402/v1/nim-sum","summary":"The nim-sum of a and b, their bitwise exclusive or. 13 xor 7 is 10, and 255 xor 256 is 511. Both run from 0 to 999999999","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mex","url":"https://scorptech.it.com/api/x402/v1/mex","summary":"The mex of a list: the smallest missing nonnegative integer. The mex of 0, 1, 3 is 2. The mex of 1, 2, 3 is 0. The mex o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cayley","url":"https://scorptech.it.com/api/x402/v1/cayley","summary":"The number of trees on n labeled vertices, n^(n-2). One vertex gives 1, two give 1, three give 3, and four give 16. n ru","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cayley-rooted","url":"https://scorptech.it.com/api/x402/v1/cayley-rooted","summary":"The number of rooted trees on n labeled vertices, n^(n-1). Four vertices give 64. n runs from 1 to 30. Past 15 digits th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prufer-encode","url":"https://scorptech.it.com/api/x402/v1/prufer-encode","summary":"The Prüfer code of a tree on vertices 1..n. Edges are a flat list of pairs. The path 1-2-3-4 gives 2,3. The star with ce","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prufer-decode","url":"https://scorptech.it.com/api/x402/v1/prufer-decode","summary":"The tree edges for a Prüfer code, as a flat list of sorted pairs. Code 2,3 is the path 1-2-3-4. Code 1,1 is the star at ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prufer-degree","url":"https://scorptech.it.com/api/x402/v1/prufer-degree","summary":"The degree of a vertex in the tree described by a Prüfer code. Degree is one plus the number of times the vertex appears","price":"$0.001","params":[]},{"method":"POST","path":"/v1/polygonal","url":"https://scorptech.it.com/api/x402/v1/polygonal","summary":"The nth s-gonal number, n*((s-2)*n - (s-4))/2. Triangular 4 is 10, square 5 is 25, pentagonal 5 is 35, and hexagonal 5 i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/polygonal-index","url":"https://scorptech.it.com/api/x402/v1/polygonal-index","summary":"The index n of an s-gonal number. Triangular 10 is index 4, and square 25 is index 5. 0 is index 0. A value that is not ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/centered-polygonal","url":"https://scorptech.it.com/api/x402/v1/centered-polygonal","summary":"The nth centered s-gonal number, s*n*(n-1)/2 + 1. Centered triangular 4 is 19, centered square 2 is 5, and centered hexa","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pronic","url":"https://scorptech.it.com/api/x402/v1/pronic","summary":"The nth pronic number, n*(n+1). The 0th is 0, the 1st is 2, and the 5th is 30. n runs from 0 to 10000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/star-number","url":"https://scorptech.it.com/api/x402/v1/star-number","summary":"The nth star number, 6*n*(n-1) + 1. The 1st is 1, the 2nd is 13, and the 5th is 121. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/twin-prime","url":"https://scorptech.it.com/api/x402/v1/twin-prime","summary":"Whether n and n+2 are both prime. 3 pairs with 5, 5 pairs with 7, and 7 does not pair with 9. n runs from 1 to 100000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cousin-prime","url":"https://scorptech.it.com/api/x402/v1/cousin-prime","summary":"Whether n and n+4 are both prime. 3 pairs with 7, and 7 pairs with 11. 5 does not pair with 9. n runs from 1 to 10000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sexy-prime","url":"https://scorptech.it.com/api/x402/v1/sexy-prime","summary":"Whether n and n+6 are both prime. 5 pairs with 11, 7 pairs with 13, and 3 does not pair with 9. n runs from 1 to 1000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sophie-germain","url":"https://scorptech.it.com/api/x402/v1/sophie-germain","summary":"Whether n and 2*n+1 are both prime. 2 gives the safeprime 5, 11 gives 23, and 7 gives 15, which is not prime. n runs fro","price":"$0.001","params":[]},{"method":"POST","path":"/v1/safeprime","url":"https://scorptech.it.com/api/x402/v1/safeprime","summary":"Whether n and (n-1)/2 are both prime. 5 comes from 2, 7 comes from 3, and 23 comes from 11. 3 and 17 are not safeprimes.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/goldbach","url":"https://scorptech.it.com/api/x402/v1/goldbach","summary":"The Goldbach split of an even n with the smallest prime p, so n = p+q and p <= q. 10 gives 3 and 7. 20 gives 3 and 17. 4","price":"$0.001","params":[]},{"method":"POST","path":"/v1/goldbach-count","url":"https://scorptech.it.com/api/x402/v1/goldbach-count","summary":"How many Goldbach splits an even n has, with p <= q. 10 has 2, 20 has 2, and 100 has 6. n is even, from 4 to 10000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/goldbach-check","url":"https://scorptech.it.com/api/x402/v1/goldbach-check","summary":"Whether p and n-p are both prime. For n=10, p=3 and p=5 work, and p=2 does not. n is even, from 4 to 10000, and p runs f","price":"$0.001","params":[]},{"method":"POST","path":"/v1/levy","url":"https://scorptech.it.com/api/x402/v1/levy","summary":"Levy writes an odd n as p+2q with p and q prime, choosing the smallest p. 7 is 3+2*2, 27 is 5+2*11, and 21 is 7+2*7. n i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/odd-goldbach","url":"https://scorptech.it.com/api/x402/v1/odd-goldbach","summary":"The lexicographically smallest triple of primes adding to an odd n. 7 is 2+2+3, 9 is 2+2+5, and 27 is 2+2+23. n is odd, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kaprekar","url":"https://scorptech.it.com/api/x402/v1/kaprekar","summary":"Whether n is a Kaprekar number. Split n^2 so the right part has as many digits as n; the two parts add to n, and the rig","price":"$0.001","params":[]},{"method":"POST","path":"/v1/automorphic","url":"https://scorptech.it.com/api/x402/v1/automorphic","summary":"Whether n^2 ends with n. 1, 5, 6, 25, 76, 376, 625, and 9376 do. 10 does not. n runs from 1 to 10000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trimorphic","url":"https://scorptech.it.com/api/x402/v1/trimorphic","summary":"Whether n^3 ends with n. 1, 4, 5, 6, 9, 25, 49, 76, 125, and 999 do. 10 does not. n runs from 1 to 10000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/narcissistic","url":"https://scorptech.it.com/api/x402/v1/narcissistic","summary":"Whether n equals the sum of its digits each raised to the number of digits. 153, 370, 371, and 407 do, and so does every","price":"$0.001","params":[]},{"method":"POST","path":"/v1/dudeney","url":"https://scorptech.it.com/api/x402/v1/dudeney","summary":"Whether n is a perfect cube and its digits add to the cube root. 1, 512, 4913, 5832, 17576, and 19683 are. 100 is not a ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factorion","url":"https://scorptech.it.com/api/x402/v1/factorion","summary":"Whether n equals the sum of the factorials of its digits. 1, 2, 145, and 40585 do. 24 and 10 do not. n runs from 1 to 10","price":"$0.001","params":[]},{"method":"POST","path":"/v1/smith","url":"https://scorptech.it.com/api/x402/v1/smith","summary":"Whether n is a composite Smith number: its digit sum equals the digit sum of its prime factors with multiplicity. 4, 22,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hoax","url":"https://scorptech.it.com/api/x402/v1/hoax","summary":"Whether n is a composite hoax number: its digit sum equals the digit sum of its distinct prime factors. 22, 58, and 84 a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sphenic","url":"https://scorptech.it.com/api/x402/v1/sphenic","summary":"Whether n is a product of three distinct primes. 30, 42, 66, and 70 are. 12 and 60 are not, because of the square factor","price":"$0.001","params":[]},{"method":"POST","path":"/v1/almost-perfect","url":"https://scorptech.it.com/api/x402/v1/almost-perfect","summary":"Whether sigma(n) = 2*n - 1. Powers of two, including 1, are almost perfect. 16 gives sigma 31. 6 and 10 are not. n runs ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mersenne","url":"https://scorptech.it.com/api/x402/v1/mersenne","summary":"The Mersenne number 2^n - 1. Exponent 1 gives 1, 2 gives 3, and 31 gives 2147483647. n runs from 1 to 128. Past 15 digit","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mersenne-prime","url":"https://scorptech.it.com/api/x402/v1/mersenne-prime","summary":"Whether 2^n - 1 is prime. Exponents 2, 3, 5, 7, 13, 17, 19, and 31 are, inside this cap. Exponent 11 is not, because 204","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fermat-number","url":"https://scorptech.it.com/api/x402/v1/fermat-number","summary":"The Fermat number 2^(2^n) + 1. n=0 gives 3, n=4 gives 65537, and n=5 gives 4294967297. n runs from 0 to 12. Past 15 digi","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fermat-prime","url":"https://scorptech.it.com/api/x402/v1/fermat-prime","summary":"Whether the Fermat number 2^(2^n) + 1 is prime. n from 0 through 4 gives a prime. n=5 does not: 4294967297 = 641*6700417","price":"$0.001","params":[]},{"method":"POST","path":"/v1/proth","url":"https://scorptech.it.com/api/x402/v1/proth","summary":"The Proth number k*2^n + 1. k must be odd and strictly less than 2^n. 1*2^1+1 is 3, and 3*2^5+1 is 97. An even k, or a k","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wilson-quotient","url":"https://scorptech.it.com/api/x402/v1/wilson-quotient","summary":"The Wilson quotient ((p-1)! + 1) / p for a prime p. The quotient of 5 is 5, of 7 is 103, and of 13 is 36846277. A compos","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wilson-prime","url":"https://scorptech.it.com/api/x402/v1/wilson-prime","summary":"Whether p^2 divides (p-1)! + 1. Inside this cap, 5 and 13 do, and 2, 3, 7, 11, and 17 do not. A composite is rejected. p","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factorial-prime","url":"https://scorptech.it.com/api/x402/v1/factorial-prime","summary":"Whether n! + 1 and n! - 1 are prime. Both are prime for n=3. For n=11, only n! + 1 is prime. For n=12, only n! - 1 is pr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/primorial-prime","url":"https://scorptech.it.com/api/x402/v1/primorial-prime","summary":"Whether n# + 1 and n# - 1 are prime, where n# is the product of the primes up to n. 5# is 30, and both neighbors are pri","price":"$0.001","params":[]},{"method":"POST","path":"/v1/brocard","url":"https://scorptech.it.com/api/x402/v1/brocard","summary":"Whether n! + 1 is a perfect square. 4! + 1 = 5^2, 5! + 1 = 11^2, and 7! + 1 = 71^2. The root is 0 when it is not a squar","price":"$0.001","params":[]},{"method":"POST","path":"/v1/square-sum","url":"https://scorptech.it.com/api/x402/v1/square-sum","summary":"The sum of the first n squares, n*(n+1)*(2n+1)/6. The sum through 1 is 1, and through 10 is 385. n runs from 0 to 100000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cube-sum","url":"https://scorptech.it.com/api/x402/v1/cube-sum","summary":"The sum of the first n cubes, (n*(n+1)/2)^2. The sum through 2 is 9, and through 10 is 3025. n runs from 0 to 100000. Pa","price":"$0.001","params":[]},{"method":"POST","path":"/v1/tetrahedral","url":"https://scorptech.it.com/api/x402/v1/tetrahedral","summary":"The nth tetrahedral number, n*(n+1)*(n+2)/6. The 5th is 35 and the 8th is 120. n runs from 0 to 1000000. Past 15 digits ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lazy-caterer","url":"https://scorptech.it.com/api/x402/v1/lazy-caterer","summary":"The lazy caterer number, the most pieces of a pancake cut by n straight cuts: n*(n+1)/2 + 1. Zero cuts give 1, three giv","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cake-number","url":"https://scorptech.it.com/api/x402/v1/cake-number","summary":"The cake number, the most pieces of space cut by n planes: (n^3 + 5*n + 6)/6. Zero planes give 1, three give 8, and five","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cullen","url":"https://scorptech.it.com/api/x402/v1/cullen","summary":"The nth Cullen number, n*2^n + 1. The 1st is 3, the 2nd is 9, and the 5th is 161. n runs from 1 to 256. Past 15 digits t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/woodall","url":"https://scorptech.it.com/api/x402/v1/woodall","summary":"The nth Woodall number, n*2^n - 1. The 1st is 1, the 2nd is 7, and the 3rd is 23. n runs from 1 to 256. Past 15 digits t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/thabit","url":"https://scorptech.it.com/api/x402/v1/thabit","summary":"The nth Thabit number, 3*2^n - 1. n=0 gives 2, n=1 gives 5, and n=2 gives 11. n runs from 0 to 256. Past 15 digits the v","price":"$0.001","params":[]},{"method":"POST","path":"/v1/carol","url":"https://scorptech.it.com/api/x402/v1/carol","summary":"The nth Carol number, (2^n - 1)^2 - 2. n=1 gives -1, n=2 gives 7, and n=3 gives 47. n runs from 1 to 128. Past 15 digits","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kynea","url":"https://scorptech.it.com/api/x402/v1/kynea","summary":"The nth Kynea number, (2^n + 1)^2 - 2. n=1 gives 7, n=2 gives 23, and n=3 gives 79. n runs from 1 to 128. Past 15 digits","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hyperperfect","url":"https://scorptech.it.com/api/x402/v1/hyperperfect","summary":"Whether n = 1 + k*(sigma(n) - n - 1) for a positive integer k, and that k. 6 and 28 give k=1. 21 gives k=2. 12 and 120 d","price":"$0.001","params":[]},{"method":"POST","path":"/v1/multiperfect","url":"https://scorptech.it.com/api/x402/v1/multiperfect","summary":"Whether sigma(n) is a multiple of n, with multiplicity k at least 2. 6 and 28 give k=2. 120 and 672 give k=3. 1 and 12 d","price":"$0.001","params":[]},{"method":"POST","path":"/v1/triperfect","url":"https://scorptech.it.com/api/x402/v1/triperfect","summary":"Whether sigma(n) = 3*n. 120 and 672 are triperfect. 6 and 28 are perfect, so they are not. n runs from 1 to 100000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/betrothed","url":"https://scorptech.it.com/api/x402/v1/betrothed","summary":"Whether n and m are betrothed: the sum of proper divisors of n is m+1, and the sum for m is n+1, with m different from n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ore","url":"https://scorptech.it.com/api/x402/v1/ore","summary":"Whether n is an Ore number: n times the divisor count is divisible by sigma(n), so the harmonic mean of the divisors is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/leyland","url":"https://scorptech.it.com/api/x402/v1/leyland","summary":"The Leyland number x^y + y^x. 2 and 2 give 8, 2 and 3 give 17, and 3 and 3 give 54. x and y each run from 2 to 20. Past ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wagstaff","url":"https://scorptech.it.com/api/x402/v1/wagstaff","summary":"The Wagstaff number (2^p + 1)/3 for an odd prime p. p=3 gives 3, p=5 gives 11, and p=7 gives 43. An even number or a com","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wagstaff-prime","url":"https://scorptech.it.com/api/x402/v1/wagstaff-prime","summary":"Whether the Wagstaff number (2^p + 1)/3 is prime. p=3, 5, 7, and 31 are. p=29 is not. An even number or a composite is r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/polite","url":"https://scorptech.it.com/api/x402/v1/polite","summary":"Whether n is polite: it is a sum of two or more consecutive positive integers. Every number that is not a power of two i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/polite-split","url":"https://scorptech.it.com/api/x402/v1/polite-split","summary":"The shortest sum of consecutive positive integers equal to n. 9 is 4+5, so it starts at 4 and has length 2. 6 is 1+2+3. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/totient-sum","url":"https://scorptech.it.com/api/x402/v1/totient-sum","summary":"The sum of Euler totients from 1 through n. Through 1 it is 1, through 5 it is 10, and through 10 it is 32. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/farey-count","url":"https://scorptech.it.com/api/x402/v1/farey-count","summary":"How many reduced fractions lie in the Farey sequence of order n, from 0/1 through 1/1. Order 1 has 2, order 5 has 11, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/visible-points","url":"https://scorptech.it.com/api/x402/v1/visible-points","summary":"How many lattice points (x, y) with 1 <= x,y <= n are visible from the origin, meaning gcd(x, y) = 1. The 1-square has 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/gcd-sum","url":"https://scorptech.it.com/api/x402/v1/gcd-sum","summary":"The sum of gcd(x, y) over 1 <= x,y <= n. Through 1 it is 1, through 5 it is 37, and through 10 it is 189. n runs from 1 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cototient","url":"https://scorptech.it.com/api/x402/v1/cototient","summary":"The cototient n - totient(n): how many integers from 1 to n share a factor with n. 1 gives 0, 6 gives 4, 9 gives 3, and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/josephus","url":"https://scorptech.it.com/api/x402/v1/josephus","summary":"The surviving seat in the Josephus circle. Seats are 1 through n, counting starts at 1, and every k-th remaining person ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/josephus-k2","url":"https://scorptech.it.com/api/x402/v1/josephus-k2","summary":"The Josephus survivor when every 2nd person is eliminated. Write n = 2^m + l with l < 2^m. The survivor is 2*l + 1. 5 gi","price":"$0.001","params":[]},{"method":"POST","path":"/v1/josephus-first","url":"https://scorptech.it.com/api/x402/v1/josephus-first","summary":"The first person eliminated in the Josephus circle. With 5 people and every 2nd, person 2 leaves first. With every 3rd, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/josephus-order","url":"https://scorptech.it.com/api/x402/v1/josephus-order","summary":"The elimination order in the Josephus circle, comma-separated, not including the survivor. With 5 people and every 2nd, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/josephus-place","url":"https://scorptech.it.com/api/x402/v1/josephus-place","summary":"When a seat is eliminated. Place 1 is the first person out. The survivor has place n. With 5 people and every 2nd, seat ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fourth-sum","url":"https://scorptech.it.com/api/x402/v1/fourth-sum","summary":"The sum of the first n fourth powers, n(n+1)(2n+1)(3n^2+3n-1)/30. The sum through 0 is 0, through 1 is 1, and through 10","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fifth-sum","url":"https://scorptech.it.com/api/x402/v1/fifth-sum","summary":"The sum of the first n fifth powers, n^2(n+1)^2(2n^2+2n-1)/12. The sum through 10 is 220825. n runs from 0 to 10000. Pas","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sixth-sum","url":"https://scorptech.it.com/api/x402/v1/sixth-sum","summary":"The sum of the first n sixth powers, n(n+1)(2n+1)(3n^4+6n^3-3n+1)/42. The sum through 10 is 1978405. n runs from 0 to 10","price":"$0.001","params":[]},{"method":"POST","path":"/v1/seventh-sum","url":"https://scorptech.it.com/api/x402/v1/seventh-sum","summary":"The sum of the first n seventh powers, n^2(n+1)^2(3n^4+6n^3-n^2-4n+2)/24. The sum through 10 is 18080425. n runs from 0 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/odd-square-sum","url":"https://scorptech.it.com/api/x402/v1/odd-square-sum","summary":"The sum of the first n odd squares: 1^2 + 3^2 + ... + (2n-1)^2 = n(4n^2-1)/3. One term is 1, two terms are 10, and four ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cyclotomic","url":"https://scorptech.it.com/api/x402/v1/cyclotomic","summary":"The cyclotomic value Phi_n(a) for an integer a at least 2. Phi_1(2) is 1, Phi_5(2) is 31, Phi_6(3) is 7, and Phi_1(5) is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cyclotomic-one","url":"https://scorptech.it.com/api/x402/v1/cyclotomic-one","summary":"Phi_n(1). It is 0 when n is 1. When n is a power of a prime p, it is p. Otherwise it is 1. Phi_4(1) is 2, Phi_6(1) is 1,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cyclotomic-neg","url":"https://scorptech.it.com/api/x402/v1/cyclotomic-neg","summary":"Phi_n(-1). n=1 gives -2, n=2 gives 0, n=6 gives 3, and n=10 gives 5. n runs from 1 to 120.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cyclotomic-coeff","url":"https://scorptech.it.com/api/x402/v1/cyclotomic-coeff","summary":"The coefficient of x^k in Phi_n. Phi_6 has coefficients 1, -1, 1, so k=1 is -1. The coefficient of x^7 in Phi_105 is -2.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cyclotomic-height","url":"https://scorptech.it.com/api/x402/v1/cyclotomic-height","summary":"The largest absolute coefficient of Phi_n. Phi_1 through Phi_15 have height 1. Phi_105 has height 2. n runs from 1 to 12","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sqrt-period","url":"https://scorptech.it.com/api/x402/v1/sqrt-period","summary":"The period of the continued fraction of sqrt(d). sqrt(2) has period 1, sqrt(3) has period 2, sqrt(7) has period 4, and s","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sqrt-cf","url":"https://scorptech.it.com/api/x402/v1/sqrt-cf","summary":"The continued fraction of sqrt(d): the integer part a0 and the repeating block, comma-separated. sqrt(2) is 1 with block","price":"$0.001","params":[]},{"method":"POST","path":"/v1/neg-pell","url":"https://scorptech.it.com/api/x402/v1/neg-pell","summary":"Whether x^2 - d*y^2 = -1 has a positive integer solution. It does exactly when the continued-fraction period of sqrt(d) ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sqrt-tail","url":"https://scorptech.it.com/api/x402/v1/sqrt-tail","summary":"The last term of the repeating block of sqrt(d), which is twice the integer part. sqrt(2) ends in 2, sqrt(3) ends in 2, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/neg-pell-count","url":"https://scorptech.it.com/api/x402/v1/neg-pell-count","summary":"How many nonsquares from 1 through n have an odd continued-fraction period, so the negative Pell equation is solvable. T","price":"$0.001","params":[]},{"method":"POST","path":"/v1/self-number","url":"https://scorptech.it.com/api/x402/v1/self-number","summary":"Whether n is a self number: no m has m plus its digit sum equal to n. 1, 20, and 108 are. 2, 10, and 21 are not. n runs ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/self-count","url":"https://scorptech.it.com/api/x402/v1/self-count","summary":"How many self numbers are at most n. Through 1 there is 1, through 10 there are 5, through 20 there are 6, and through 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/self-next","url":"https://scorptech.it.com/api/x402/v1/self-next","summary":"The smallest self number that is at least n. 1 stays 1, 2 gives 3, 10 gives 20, and 21 gives 31. n runs from 1 to 100000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/self-nth","url":"https://scorptech.it.com/api/x402/v1/self-nth","summary":"The kth self number. The 1st is 1, the 5th is 9, the 10th is 64, and the 20th is 165. k runs from 1 to 2000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/self-seed","url":"https://scorptech.it.com/api/x402/v1/self-seed","summary":"m plus the sum of its decimal digits. 1 gives 2, 5 gives 10, 19 gives 29, and 99 gives 117. A self number is a number th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factoradic","url":"https://scorptech.it.com/api/x402/v1/factoradic","summary":"The factorial-base digits of n, from the highest place down through the 0! place, comma-separated. The last digit is alw","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factoradic-decode","url":"https://scorptech.it.com/api/x402/v1/factoradic-decode","summary":"The integer value of factorial-base digits. The last entry is the 0! place. [2, 1, 0] is 5 and [3, 2, 1, 0] is 23. A dig","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factoradic-digit","url":"https://scorptech.it.com/api/x402/v1/factoradic-digit","summary":"The digit of the k! place of n. It is floor(n / k!) mod (k+1), and it is 0 when k! exceeds n. The 0! digit is always 0. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factoradic-length","url":"https://scorptech.it.com/api/x402/v1/factoradic-length","summary":"How many factorial-base digits n has, including the 0! place. 0 has 1, 5 has 3, 23 has 4, and 24 has 5. n runs from 0 to","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factoradic-sum","url":"https://scorptech.it.com/api/x402/v1/factoradic-sum","summary":"The sum of the factorial-base digits of n. 0 sums to 0, 5 sums to 3, 23 sums to 6, and 463 sums to 9. n runs from 0 to 9","price":"$0.001","params":[]},{"method":"POST","path":"/v1/binary-partition","url":"https://scorptech.it.com/api/x402/v1/binary-partition","summary":"The number of binary partitions of n, partitions into powers of two. b(0) is 1. Odd n copies b(n-1). Even n is b(n-1) pl","price":"$0.001","params":[]},{"method":"POST","path":"/v1/binary-partition-sum","url":"https://scorptech.it.com/api/x402/v1/binary-partition-sum","summary":"The sum of the binary partition numbers from b(0) through b(n). Through 0 it is 1, through 4 it is 10, through 10 it is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/plane-partition","url":"https://scorptech.it.com/api/x402/v1/plane-partition","summary":"The number of plane partitions of n. The values begin 1, 1, 3, 6, 13, 24, so n=5 is 24, n=9 is 282, and n=10 is 500. n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/plane-partition-sum","url":"https://scorptech.it.com/api/x402/v1/plane-partition-sum","summary":"The sum of the plane partition numbers from 0 through n. Through 0 it is 1, through 2 it is 5, through 4 it is 24, and t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/overpartition","url":"https://scorptech.it.com/api/x402/v1/overpartition","summary":"The number of overpartitions of n. The first part may be overlined. The values begin 1, 2, 4, 8, 14, 24, so n=4 is 14 an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/highly-abundant","url":"https://scorptech.it.com/api/x402/v1/highly-abundant","summary":"Whether sigma(n) is strictly larger than sigma(m) for every m below n. 1, 6, 8, and 16 are. 5 and 9 are not. n runs from","price":"$0.001","params":[]},{"method":"POST","path":"/v1/superabundant","url":"https://scorptech.it.com/api/x402/v1/superabundant","summary":"Whether sigma(n)/n is strictly larger than sigma(m)/m for every m below n. The ratio is compared as sigma(n)*m against s","price":"$0.001","params":[]},{"method":"POST","path":"/v1/primitive-abundant","url":"https://scorptech.it.com/api/x402/v1/primitive-abundant","summary":"Whether n is abundant and no proper divisor is abundant. 12, 18, 20, and 70 are. 24 is not, because 12 divides it. 6 is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/abundance","url":"https://scorptech.it.com/api/x402/v1/abundance","summary":"sigma(n) minus 2n. A perfect number gives 0, an abundant number gives a positive value, and a deficient number gives a n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/perfect-totient","url":"https://scorptech.it.com/api/x402/v1/perfect-totient","summary":"Whether n equals the sum of its iterated totients down to 1. The sum includes the final 1 and excludes n. 3, 9, 15, and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/balanced-prime","url":"https://scorptech.it.com/api/x402/v1/balanced-prime","summary":"Whether n is prime and equal to the average of the previous prime and the next prime. 5, 53, 157, and 1103 are. 2, 3, 7,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prime-triplet","url":"https://scorptech.it.com/api/x402/v1/prime-triplet","summary":"Whether n starts a prime triplet: n, n+2, and n+6 are prime, or n, n+4, and n+6 are prime. 5, 7, 11, and 13 do. 3 and 19","price":"$0.001","params":[]},{"method":"POST","path":"/v1/prime-triplet-count","url":"https://scorptech.it.com/api/x402/v1/prime-triplet-count","summary":"How many prime-triplet starters are at most n. Through 3 there are none, through 5 there is 1, through 13 there are 4, a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/chen-prime","url":"https://scorptech.it.com/api/x402/v1/chen-prime","summary":"Whether n is prime and n+2 is either prime or a product of two primes, counted with multiplicity. 2, 41, and 47 are. 43 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/depolignac","url":"https://scorptech.it.com/api/x402/v1/depolignac","summary":"Whether odd n is never a prime plus a power of two, counting 2^0. 1 qualifies, because nothing smaller works. 3 is 2+1, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/arithmetic-derivative","url":"https://scorptech.it.com/api/x402/v1/arithmetic-derivative","summary":"The arithmetic derivative of n. 0 and 1 give 0, a prime gives 1, 4 gives 4, 6 gives 5, 8 gives 12, and 12 gives 16. n ru","price":"$0.001","params":[]},{"method":"POST","path":"/v1/arithmetic-second","url":"https://scorptech.it.com/api/x402/v1/arithmetic-second","summary":"The second arithmetic derivative of n. 6 goes 5 then 1. 8 goes 12 then 16. 4 stays 4. 0 and 1 stay 0. n runs from 0 to 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/derivative-fixed","url":"https://scorptech.it.com/api/x402/v1/derivative-fixed","summary":"Whether n equals its arithmetic derivative. 0, 4, 27, and 3125 do. 1, 6, and 16 do not. n runs from 0 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sopf","url":"https://scorptech.it.com/api/x402/v1/sopf","summary":"The sum of the distinct prime factors of n. 1 gives 0, 8 gives 2, 12 gives 5, and 30 gives 10. A prime gives itself. n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ruth-aaron","url":"https://scorptech.it.com/api/x402/v1/ruth-aaron","summary":"Whether n and n+1 have the same sum of prime factors, counted with repetition. 5, 8, 15, and 714 do. 6 and 9 do not. n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mcnugget","url":"https://scorptech.it.com/api/x402/v1/mcnugget","summary":"Whether n nuggets can be ordered in boxes of 6, 9, and 20. 0, 6, and 44 can. 1, 11, and 43 cannot. n runs from 0 to 5000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mcnugget-ways","url":"https://scorptech.it.com/api/x402/v1/mcnugget-ways","summary":"How many combinations of 6, 9, and 20 boxes add to n. 0 has 1 empty order, 6 has 1, 18 has 2, 43 has 0, and 44 has 2. n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mcnugget-min","url":"https://scorptech.it.com/api/x402/v1/mcnugget-min","summary":"The fewest boxes of 6, 9, and 20 that add to n. 0 takes 0, 6 takes 1, 12 takes 2, and 44 takes 4. An amount that cannot ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mcnugget-count","url":"https://scorptech.it.com/api/x402/v1/mcnugget-count","summary":"How many integers from 0 through n can be ordered in boxes of 6, 9, and 20. Through 0 there is 1, through 6 there are 2,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/mcnugget-miss","url":"https://scorptech.it.com/api/x402/v1/mcnugget-miss","summary":"How many positive integers through n cannot be ordered in boxes of 6, 9, and 20. Through 1 there is 1, through 6 there a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-steps","url":"https://scorptech.it.com/api/x402/v1/collatz-steps","summary":"How many Collatz steps it takes to reach 1. 1 takes 0, 2 takes 1, 3 takes 7, 6 takes 8, 7 takes 16, and 27 takes 111. n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-max","url":"https://scorptech.it.com/api/x402/v1/collatz-max","summary":"The largest value on the Collatz path from n through 1, including n. 1 peaks at 1, 3 peaks at 16, 7 peaks at 52, and 27 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-next","url":"https://scorptech.it.com/api/x402/v1/collatz-next","summary":"One Collatz step. An even number is halved and an odd number goes to 3n+1. 1 goes to 4, 2 goes to 1, 3 goes to 10, and 6","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-drop","url":"https://scorptech.it.com/api/x402/v1/collatz-drop","summary":"How many Collatz steps until the value is strictly below the start. 1 is already at the bottom and takes 0. 2 takes 1, 6","price":"$0.001","params":[]},{"method":"POST","path":"/v1/collatz-odd","url":"https://scorptech.it.com/api/x402/v1/collatz-odd","summary":"How many odd terms appear before the Collatz path reaches 1. The final 1 is not counted. 1 and 2 give 0, 3 and 6 give 2,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/squarefree-part","url":"https://scorptech.it.com/api/x402/v1/squarefree-part","summary":"n divided by its largest square divisor. 12 gives 3, 8 gives 2, 18 gives 2, 36 gives 1, and a square-free n gives itself","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cubefree","url":"https://scorptech.it.com/api/x402/v1/cubefree","summary":"Whether no cube divides n. 1, 12, 18, and 36 are cubefree. 8, 16, 24, and 72 are not. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fourth-free","url":"https://scorptech.it.com/api/x402/v1/fourth-free","summary":"Whether no fourth power divides n. 1, 8, 12, and 36 qualify. 16, 32, 48, and 81 do not. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/squarefree-count","url":"https://scorptech.it.com/api/x402/v1/squarefree-count","summary":"How many square-free positive integers are at most n. Through 1 there is 1, through 4 there are 3, through 10 there are ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cubefree-count","url":"https://scorptech.it.com/api/x402/v1/cubefree-count","summary":"How many cubefree positive integers are at most n. Through 7 there are 7, through 8 there are still 7, through 16 there ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wythoff-lower","url":"https://scorptech.it.com/api/x402/v1/wythoff-lower","summary":"The lower Wythoff number floor(n*phi), phi the golden ratio. n=1 gives 1, n=2 gives 3, n=5 gives 8, and n=10 gives 16. n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wythoff-kind","url":"https://scorptech.it.com/api/x402/v1/wythoff-kind","summary":"Which Wythoff sequence contains m, and the index. 1 is lower index 1. 2 is upper index 1. 4 is lower index 3. 5 is upper","price":"$0.001","params":[]},{"method":"POST","path":"/v1/wythoff-sum","url":"https://scorptech.it.com/api/x402/v1/wythoff-sum","summary":"The sum of the lower Wythoff numbers from index 1 through n. Through 1 it is 1, through 5 it is 22, and through 10 it is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/beatty-sqrt2","url":"https://scorptech.it.com/api/x402/v1/beatty-sqrt2","summary":"floor(n*sqrt(2)). n=1 gives 1, n=2 gives 2, n=3 gives 4, n=5 gives 7, and n=10 gives 14. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/beatty-sqrt2-comp","url":"https://scorptech.it.com/api/x402/v1/beatty-sqrt2-comp","summary":"The Beatty complement of sqrt(2), floor(n*(2+sqrt(2))). n=1 gives 3, n=2 gives 6, n=3 gives 10, and n=5 gives 17. n runs","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moser","url":"https://scorptech.it.com/api/x402/v1/moser","summary":"The nth Moser-de Bruijn number, binary digits of n read in base 4. n=0 gives 0, n=3 gives 5, n=5 gives 17, n=6 gives 20,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moser-sum","url":"https://scorptech.it.com/api/x402/v1/moser-sum","summary":"The sum of the Moser-de Bruijn numbers a(0) through a(n). Through 0 it is 0, through 3 it is 10, through 5 it is 43, and","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moser-check","url":"https://scorptech.it.com/api/x402/v1/moser-check","summary":"Whether every base-4 digit of n is 0 or 1. 0, 1, 4, 5, and 20 qualify. 2 and 6 do not. n runs from 0 to 1000000000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moser-index","url":"https://scorptech.it.com/api/x402/v1/moser-index","summary":"The index whose Moser-de Bruijn number is n. 0 is index 0, 5 is index 3, 20 is index 6, and 64 is index 8. A base-4 digi","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moser-count","url":"https://scorptech.it.com/api/x402/v1/moser-count","summary":"How many Moser-de Bruijn numbers are at most n, counting 0. Through 0 there is 1, through 1 there are 2, through 5 there","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hofstadter","url":"https://scorptech.it.com/api/x402/v1/hofstadter","summary":"Hofstadter G. G(0) is 0, G(1) and G(2) are 1, G(5) is 3, and G(10) is 6. The rule is G(n) = n - G(G(n-1)). n runs from 0","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hofstadter-sum","url":"https://scorptech.it.com/api/x402/v1/hofstadter-sum","summary":"The sum of Hofstadter G from 0 through n. Through 0 it is 0, through 1 it is 1, through 5 it is 10, and through 10 it is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hofstadter-f","url":"https://scorptech.it.com/api/x402/v1/hofstadter-f","summary":"Hofstadter's female sequence. F(0) and F(1) are 1, F(5) is 3, and F(7) and F(8) are 5. The male term at the same index i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hofstadter-m","url":"https://scorptech.it.com/api/x402/v1/hofstadter-m","summary":"Hofstadter's male sequence. M(0) and M(1) are 0, M(2) is 1, M(5) is 3, and M(7) is 4. n runs from 0 to 5000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hofstadter-conway","url":"https://scorptech.it.com/api/x402/v1/hofstadter-conway","summary":"Conway's Hofstadter sequence. a(1) and a(2) are 1, a(6) is 4, a(10) is 6, and a(16) is 8. The rule is a(n) = a(a(n-1)) +","price":"$0.001","params":[]},{"method":"POST","path":"/v1/euclid-mullin","url":"https://scorptech.it.com/api/x402/v1/euclid-mullin","summary":"The nth Euclid-Mullin prime. a(1) is 2, a(4) is 43, a(7) is 5, and a(8) is 6221671. n runs from 1 to 8.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/euclid-mullin-prod","url":"https://scorptech.it.com/api/x402/v1/euclid-mullin-prod","summary":"The product of the first n Euclid-Mullin primes. n=1 gives 2, n=4 gives 1806, n=7 gives 6221670, and n=8 gives 387091838","price":"$0.001","params":[]},{"method":"POST","path":"/v1/euclid-mullin-sum","url":"https://scorptech.it.com/api/x402/v1/euclid-mullin-sum","summary":"The sum of the first n Euclid-Mullin primes. Through 1 it is 2, through 4 it is 55, through 7 it is 126, and through 8 i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/euclid-mullin-cofactor","url":"https://scorptech.it.com/api/x402/v1/euclid-mullin-cofactor","summary":"One more than the product of the earlier terms, divided by a(n). It is 1 when that Euclid number is itself prime. n=5 gi","price":"$0.001","params":[]},{"method":"POST","path":"/v1/euclid-mullin-index","url":"https://scorptech.it.com/api/x402/v1/euclid-mullin-index","summary":"Where p sits among the first eight Euclid-Mullin primes, or 0 if it does not. 2 is 1, 43 is 4, 5 is 7, and 11 is 0. p ru","price":"$0.001","params":[]},{"method":"POST","path":"/v1/e-cf-term","url":"https://scorptech.it.com/api/x402/v1/e-cf-term","summary":"The nth partial quotient of e, starting at n=0. It is 2, then 1, 2, 1, 1, 4, 1, 1, 6. n=0 gives 2, n=5 gives 4, and n=8 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/e-cf-sum","url":"https://scorptech.it.com/api/x402/v1/e-cf-sum","summary":"The sum of the partial quotients of e from index 0 through n. Through 0 it is 2, through 2 it is 5, through 5 it is 11, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/e-cf-prefix","url":"https://scorptech.it.com/api/x402/v1/e-cf-prefix","summary":"The partial quotients of e from index 0 through n, comma-separated. Through 0 it is \"2\". Through 5 it is \"2,1,2,1,1,4\". ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/e-convergent","url":"https://scorptech.it.com/api/x402/v1/e-convergent","summary":"The nth convergent of e, as numerator and denominator. n=0 is 2/1, n=1 is 3/1, n=2 is 8/3, and n=5 is 87/32. n runs from","price":"$0.001","params":[]},{"method":"POST","path":"/v1/e-series","url":"https://scorptech.it.com/api/x402/v1/e-series","summary":"The partial sum of 1/k! from k=0 through n, as a reduced fraction. n=0 is 1/1, n=2 is 5/2, n=3 is 8/3, n=4 is 65/24, and","price":"$0.001","params":[]},{"method":"POST","path":"/v1/keith","url":"https://scorptech.it.com/api/x402/v1/keith","summary":"Whether n is a Keith number. 14, 19, 28, 47, 61, 75, and 197 are. 10, 11, and any one-digit number are not. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/keith-row","url":"https://scorptech.it.com/api/x402/v1/keith-row","summary":"The digit recurrence of a Keith number, comma-separated, through n itself. 14 gives \"1,4,5,9,14\". A number that is not K","price":"$0.001","params":[]},{"method":"POST","path":"/v1/keith-steps","url":"https://scorptech.it.com/api/x402/v1/keith-steps","summary":"How many new terms are generated after the digits before a Keith number reaches itself. 14 takes 3 and 197 takes 5. A nu","price":"$0.001","params":[]},{"method":"POST","path":"/v1/keith-count","url":"https://scorptech.it.com/api/x402/v1/keith-count","summary":"How many Keith numbers are at most n. Through 13 there are none, through 14 there is 1, through 100 there are 6, and thr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/keith-next","url":"https://scorptech.it.com/api/x402/v1/keith-next","summary":"The smallest Keith number that is at least n. 1 and 14 give 14, 15 gives 19, and 76 gives 197. n runs from 1 to 2000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cube-min","url":"https://scorptech.it.com/api/x402/v1/cube-min","summary":"The fewest positive cubes that add to n. 0 takes 0, 1 takes 1, 8 takes 1, 9 takes 2, 23 takes 9, and 29 takes 3. n runs ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cube-ways","url":"https://scorptech.it.com/api/x402/v1/cube-ways","summary":"How many ways n is a sum of positive cubes, ignoring order and allowing repeats. 0 has 1 empty sum, 2 has 1, 8 has 2, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/two-cubes","url":"https://scorptech.it.com/api/x402/v1/two-cubes","summary":"How many ways n is a sum of two positive cubes, ignoring order and allowing the two cubes to be equal. 2 has 1, 9 has 1,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/taxicab","url":"https://scorptech.it.com/api/x402/v1/taxicab","summary":"Whether n has at least two different writings as a sum of two positive cubes. 1729 and 4104 do. 91 and 100 do not. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/taxicab-count","url":"https://scorptech.it.com/api/x402/v1/taxicab-count","summary":"How many positive integers through n have at least two writings as a sum of two positive cubes. Through 100 there are no","price":"$0.001","params":[]},{"method":"POST","path":"/v1/palindrome","url":"https://scorptech.it.com/api/x402/v1/palindrome","summary":"Whether n reads the same forwards and backwards. 0, 7, and 121 do. 10 and 89 do not. n runs from 0 to 999999999999999.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/reverse-add","url":"https://scorptech.it.com/api/x402/v1/reverse-add","summary":"n plus its digit reversal. 10 gives 11, 19 gives 110, and 121 gives 242. A trailing zero reverses as a leading zero and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/palindrome-steps","url":"https://scorptech.it.com/api/x402/v1/palindrome-steps","summary":"How many reverse-and-add steps turn n into a palindrome. 7 and 121 take 0, 10 takes 1, 19 takes 2, and 89 takes 24. If 5","price":"$0.001","params":[]},{"method":"POST","path":"/v1/palindrome-reach","url":"https://scorptech.it.com/api/x402/v1/palindrome-reach","summary":"The palindrome reached by reverse-and-add. 10 reaches 11, 19 reaches 121, and 89 reaches 8813200023188. A palindrome rea","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lychrel","url":"https://scorptech.it.com/api/x402/v1/lychrel","summary":"Whether 50 reverse-and-add steps fail to reach a palindrome. 196 and 295 do. 89, 121, and 10 do not. n runs from 0 to 10","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kaprekar-step","url":"https://scorptech.it.com/api/x402/v1/kaprekar-step","summary":"One step of the four-digit Kaprekar routine, padding with leading zeros. 3524 gives 3087, 14 gives 4086, 6174 gives 6174","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kaprekar-steps","url":"https://scorptech.it.com/api/x402/v1/kaprekar-steps","summary":"How many steps the four-digit Kaprekar routine takes to reach 6174. 6174 takes 0, 3524 takes 3, 5 takes 6, and 14 takes ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kaprekar-chain","url":"https://scorptech.it.com/api/x402/v1/kaprekar-chain","summary":"The four-digit Kaprekar chain from n through 6174, comma-separated. 3524 gives \"3524,3087,8352,6174\" and 6174 gives \"617","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kaprekar-repdigit","url":"https://scorptech.it.com/api/x402/v1/kaprekar-repdigit","summary":"Whether the four-digit padding of n uses only one digit. 0, 1111, and 2222 do. 5, 111, and 6174 do not. n runs from 0 to","price":"$0.001","params":[]},{"method":"POST","path":"/v1/kaprekar-kernel","url":"https://scorptech.it.com/api/x402/v1/kaprekar-kernel","summary":"The value the four-digit Kaprekar routine reaches. Every non-repdigit from 0 to 9999 reaches 6174. A repdigit is rejecte","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factor-digits","url":"https://scorptech.it.com/api/x402/v1/factor-digits","summary":"How many digits the written prime factorization uses, omitting exponents of 1. 1 uses 1, 4 uses 2, 12 uses 3, 16 uses 2,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/frugal","url":"https://scorptech.it.com/api/x402/v1/frugal","summary":"Whether the written factorization uses fewer digits than n. 125 and 128 do. 1, 16, and 12 do not. n runs from 1 to 10000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/equidigital","url":"https://scorptech.it.com/api/x402/v1/equidigital","summary":"Whether the written factorization uses exactly as many digits as n. 1, 10, 16, and 25 do. 4 and 125 do not. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/extravagant","url":"https://scorptech.it.com/api/x402/v1/extravagant","summary":"Whether the written factorization uses more digits than n. 4, 6, and 12 do. 10 and 125 do not. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/frugal-count","url":"https://scorptech.it.com/api/x402/v1/frugal-count","summary":"How many frugal numbers are at most n. Through 124 there are none, through 125 there is 1, and through 200 there are 2. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zuckerman","url":"https://scorptech.it.com/api/x402/v1/zuckerman","summary":"Whether n has no zero digit and the product of its digits divides n. 1, 11, 12, 24, and 36 do. 10, 13, and 22 do not. n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zuckerman-count","url":"https://scorptech.it.com/api/x402/v1/zuckerman-count","summary":"How many Zuckerman numbers are at most n. Through 9 there are 9, through 12 there are 11, through 24 there are 13, and t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/lynch-bell","url":"https://scorptech.it.com/api/x402/v1/lynch-bell","summary":"Whether every digit of n is nonzero, the digits are all different, and each digit divides n. 12, 15, 24, 124, and 135 do","price":"$0.001","params":[]},{"method":"POST","path":"/v1/polydivisible","url":"https://scorptech.it.com/api/x402/v1/polydivisible","summary":"Whether the first k digits of n form a multiple of k, for every k. 0, 10, 12, 24, 102, and 3816547290 do. 11 and 103 do ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/polydivisible-count","url":"https://scorptech.it.com/api/x402/v1/polydivisible-count","summary":"How many polydivisible numbers from 0 through n there are. Through 1 there are 2, through 10 there are 11, through 12 th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/vampire","url":"https://scorptech.it.com/api/x402/v1/vampire","summary":"Whether n is a vampire number. 1260, 1395, 1435, and 6880 are. 1000 and 1261 are not. n runs from 1 to 9999.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/vampire-fangs","url":"https://scorptech.it.com/api/x402/v1/vampire-fangs","summary":"The smaller-first fang pair of a vampire number, comma-separated. 1260 gives \"21,60\" and 6880 gives \"80,86\". A non-vampi","price":"$0.001","params":[]},{"method":"POST","path":"/v1/vampire-low","url":"https://scorptech.it.com/api/x402/v1/vampire-low","summary":"The smaller fang of a vampire number. 1260 gives 21, 1395 gives 15, and 6880 gives 80. A non-vampire is rejected. n runs","price":"$0.001","params":[]},{"method":"POST","path":"/v1/vampire-count","url":"https://scorptech.it.com/api/x402/v1/vampire-count","summary":"How many vampire numbers are at most n. Through 1259 there are none, through 1500 there are 3, through 2187 there are 6,","price":"$0.001","params":[]},{"method":"POST","path":"/v1/vampire-next","url":"https://scorptech.it.com/api/x402/v1/vampire-next","summary":"The smallest vampire number that is at least n. 1000 and 1260 give 1260, 1261 gives 1395, and 2188 gives 6880. n runs fr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/home-prime","url":"https://scorptech.it.com/api/x402/v1/home-prime","summary":"The home prime of n. 2 stays 2, 4 becomes 211, 10 becomes 773, and 16 becomes 31636373. The search stops after 20 steps ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/home-prime-steps","url":"https://scorptech.it.com/api/x402/v1/home-prime-steps","summary":"How many factor-concatenations turn n into a prime. 2 takes 0, 6 and 12 take 1, 4 takes 2, and 10 and 16 take 4. A chain","price":"$0.001","params":[]},{"method":"POST","path":"/v1/factor-concat","url":"https://scorptech.it.com/api/x402/v1/factor-concat","summary":"One concatenation of the prime factors of n, repeats included and primes left unchanged. 2 gives 2, 4 gives 22, 8 gives ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/home-prime-len","url":"https://scorptech.it.com/api/x402/v1/home-prime-len","summary":"How many digits the home prime of n has. 2 has 1, 4 and 10 have 3, and 16 has 8. A chain past 20 steps or 12 digits is r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/home-grows","url":"https://scorptech.it.com/api/x402/v1/home-grows","summary":"Whether the home prime of n has more digits than n. 4, 6, and 10 do. 2 and 21 do not. A chain past 20 steps or 12 digits","price":"$0.001","params":[]},{"method":"POST","path":"/v1/champernowne-digit","url":"https://scorptech.it.com/api/x402/v1/champernowne-digit","summary":"The digit in place pos of 123456789101112..., counting from 1. Place 9 is 9, place 10 is 1, place 11 is 0, and place 190","price":"$0.001","params":[]},{"method":"POST","path":"/v1/champernowne-owner","url":"https://scorptech.it.com/api/x402/v1/champernowne-owner","summary":"Which integer owns place pos in the Champernowne digit stream. Place 10 and 11 belong to 10, place 12 belongs to 11, and","price":"$0.001","params":[]},{"method":"POST","path":"/v1/champernowne-place","url":"https://scorptech.it.com/api/x402/v1/champernowne-place","summary":"Where place pos sits inside its integer, counting from 1 on the left. Place 10 is position 1 of 10, place 11 is position","price":"$0.001","params":[]},{"method":"POST","path":"/v1/champernowne-start","url":"https://scorptech.it.com/api/x402/v1/champernowne-start","summary":"The first Champernowne place occupied by the integer m. 1 starts at 1, 10 starts at 10, 11 starts at 12, and 100 starts ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/champernowne-len","url":"https://scorptech.it.com/api/x402/v1/champernowne-len","summary":"How many digits are used to write the integers from 1 through m. Through 9 there are 9, through 10 there are 11, through","price":"$0.001","params":[]},{"method":"POST","path":"/v1/complexity","url":"https://scorptech.it.com/api/x402/v1/complexity","summary":"The integer complexity of n, the fewest 1s added and multiplied to reach n. 1 is 1, 6 is 5, 12 is 7, and 23 is 11. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/complexity-sum","url":"https://scorptech.it.com/api/x402/v1/complexity-sum","summary":"The sum of the integer complexities from 1 through n. Through 1 it is 1, through 10 it is 45, and through 12 it is 60. n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/complexity-max","url":"https://scorptech.it.com/api/x402/v1/complexity-max","summary":"The largest integer complexity attained from 1 through n. Through 1 it is 1, through 10 it is 7, and through 12 it is 8.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/complexity-count","url":"https://scorptech.it.com/api/x402/v1/complexity-count","summary":"How many integers from 1 through n have the same complexity as n. 5 has 1, 6 has 2, 10 has 1, and 12 has 2. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/complexity-least","url":"https://scorptech.it.com/api/x402/v1/complexity-least","summary":"The smallest positive integer whose complexity is k. k=1 gives 1, k=6 gives 7, k=8 gives 11, k=11 gives 23, and k=12 giv","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zumkeller","url":"https://scorptech.it.com/api/x402/v1/zumkeller","summary":"Whether the divisors of n split into two sets with equal sum. 6, 12, 20, and 945 do. 1, 10, and 18 do not. n runs from 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zumkeller-half","url":"https://scorptech.it.com/api/x402/v1/zumkeller-half","summary":"Half the sum of the divisors of n. 6 gives 6, 10 gives 9, and 12 gives 14. An odd divisor-sum is rejected. n runs from 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zumkeller-subset","url":"https://scorptech.it.com/api/x402/v1/zumkeller-subset","summary":"One set of divisors that sums to half the divisor-sum, comma-separated, taking the earliest divisors. 6 gives \"1,2,3\" an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zumkeller-count","url":"https://scorptech.it.com/api/x402/v1/zumkeller-count","summary":"How many Zumkeller numbers are at most n. Through 1 there are none, through 6 there is 1, through 12 there are 2, throug","price":"$0.001","params":[]},{"method":"POST","path":"/v1/zumkeller-odd","url":"https://scorptech.it.com/api/x402/v1/zumkeller-odd","summary":"Whether n is an odd Zumkeller number. 945 is. 1, 9, and 15 are not, and an even Zumkeller number such as 6 is not. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/juggler-step","url":"https://scorptech.it.com/api/x402/v1/juggler-step","summary":"One juggler step. An even n goes to its integer square root, and an odd n goes to the integer square root of its cube. 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/juggler-steps","url":"https://scorptech.it.com/api/x402/v1/juggler-steps","summary":"How many juggler steps reach 1. 1 takes 0, 2 takes 1, 3 takes 6, 7 takes 4, and 9 takes 7. n runs from 1 to 20.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/juggler-max","url":"https://scorptech.it.com/api/x402/v1/juggler-max","summary":"The largest term on the juggler path from n down to 1. 1 peaks at 1, 3 peaks at 36, and 9 peaks at 140. n runs from 1 to","price":"$0.001","params":[]},{"method":"POST","path":"/v1/juggler-odd","url":"https://scorptech.it.com/api/x402/v1/juggler-odd","summary":"How many odd terms appear before the final 1 on the juggler path. 1 and 2 have none, 7 has 1, 5 has 2, and 3 and 9 have ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/juggler-chain","url":"https://scorptech.it.com/api/x402/v1/juggler-chain","summary":"The juggler path from n through 1, comma-separated. 1 gives \"1\" and 3 gives \"3,5,11,36,6,2,1\". n runs from 1 to 20.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/narayana-cows","url":"https://scorptech.it.com/api/x402/v1/narayana-cows","summary":"Narayana's cow sequence at n, counting from 1. The first three terms are 1, and each later term is the previous term plu","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cows-sum","url":"https://scorptech.it.com/api/x402/v1/cows-sum","summary":"The sum of the first n Narayana cow numbers. The first term sums to 1, the first 3 sum to 3, the first 6 sum to 12, and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cows-next","url":"https://scorptech.it.com/api/x402/v1/cows-next","summary":"The smallest Narayana cow number that is at least n. 1 gives 1, 3 gives 3, 5 gives 6, and 130 gives 189. n runs from 1 t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cows-index","url":"https://scorptech.it.com/api/x402/v1/cows-index","summary":"The first index, counting from 1, where Narayana's cow sequence equals n. 0 means n is not a term. 1 is 1, 2 is 4, 4 is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/cows-mod","url":"https://scorptech.it.com/api/x402/v1/cows-mod","summary":"Narayana's cow term a(n) modulo m. a(11) mod 5 is 3, and a(15) mod 10 is 9. n runs from 1 to 60 and m runs from 1 to 100","price":"$0.001","params":[]},{"method":"POST","path":"/v1/digit-power-sum","url":"https://scorptech.it.com/api/x402/v1/digit-power-sum","summary":"The sum of the digits of n each raised to k. 0 to any k from 1 up is 0. 153 to the 3 is 153. 4150 to the 5 is 4150, and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pdi","url":"https://scorptech.it.com/api/x402/v1/pdi","summary":"Whether n equals the sum of its digits each raised to k. 153 with k=3, 1634 with k=4, and 4150 with k=5 do. 4150 with k=","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pdi-count","url":"https://scorptech.it.com/api/x402/v1/pdi-count","summary":"How many integers from 0 through n equal the sum of their digits each raised to k. For k=3, through 1 there are 2 and th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/munchausen","url":"https://scorptech.it.com/api/x402/v1/munchausen","summary":"Whether n equals the sum of its digits each raised to itself, taking 0^0 as 1. 1 and 3435 do. 0, 2, and 438579088 do not","price":"$0.001","params":[]},{"method":"POST","path":"/v1/munchausen-sum","url":"https://scorptech.it.com/api/x402/v1/munchausen-sum","summary":"The sum of the digits of n each raised to itself, taking 0^0 as 1. 0 gives 1, 1 gives 1, 2 gives 4, 10 gives 2, 3435 giv","price":"$0.001","params":[]},{"method":"POST","path":"/v1/negabinary","url":"https://scorptech.it.com/api/x402/v1/negabinary","summary":"The base -2 digits of n, without a leading zero except for 0 itself. 0 is \"0\", 1 is \"1\", 2 is \"110\", 4 is \"100\", and 6 i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/negabinary-decode","url":"https://scorptech.it.com/api/x402/v1/negabinary-decode","summary":"The integer written by a base -2 bit string. \"0\" is 0, \"110\" is 2, \"100\" is 4, and \"11010\" is 6. A character other than ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/negabinary-digit","url":"https://scorptech.it.com/api/x402/v1/negabinary-digit","summary":"The base -2 digit of n in place, counting from 0 at the right. For 6, which is \"11010\", place 0 is 0, place 1 is 1, and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/negabinary-len","url":"https://scorptech.it.com/api/x402/v1/negabinary-len","summary":"How many digits the base -2 writing of n uses. 0 uses 1, 4 uses 3, and 6 uses 5. n runs from 0 to 999999999999.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/negabinary-bits","url":"https://scorptech.it.com/api/x402/v1/negabinary-bits","summary":"How many 1 digits the base -2 writing of n has. 0 has none, 1 has 1, 6 has 3, and 7 has 4. n runs from 0 to 999999999999","price":"$0.001","params":[]},{"method":"POST","path":"/v1/balanced-ternary","url":"https://scorptech.it.com/api/x402/v1/balanced-ternary","summary":"The balanced ternary digits of n. -1 is written as T. 0 is \"0\", 2 is \"1T\", 4 is \"11\", 5 is \"1TT\", and 6 is \"1T0\". n runs","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ternary-value","url":"https://scorptech.it.com/api/x402/v1/ternary-value","summary":"The integer written by balanced ternary digits, where T means -1. \"0\" is 0, \"T\" is -1, \"1T\" is 2, \"10\" is 3, and \"1TT\" i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ternary-digit","url":"https://scorptech.it.com/api/x402/v1/ternary-digit","summary":"The balanced ternary digit of n in place, counting from 0 at the right, as -1, 0, or 1. For 5, which is \"1TT\", place 0 a","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ternary-len","url":"https://scorptech.it.com/api/x402/v1/ternary-len","summary":"How many balanced ternary digits n uses. 0 uses 1, 5 uses 3, and 14 uses 4. n runs from 0 to 999999999999999.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/ternary-nonzero","url":"https://scorptech.it.com/api/x402/v1/ternary-nonzero","summary":"How many nonzero balanced ternary digits n uses. 0 has none, 3 has 1, 6 has 2, and 5 has 3. n runs from 0 to 99999999999","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hamming","url":"https://scorptech.it.com/api/x402/v1/hamming","summary":"Whether n is a Hamming number, that is 5-smooth. 1, 10, 30, and 36 are. 7 and 14 are not. n runs from 1 to 1000000000000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hamming-nth","url":"https://scorptech.it.com/api/x402/v1/hamming-nth","summary":"The nth Hamming number, counting from 1. The 1st is 1, the 10th is 12, the 18th is 30, and the 20th is 36. n runs from 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hamming-count","url":"https://scorptech.it.com/api/x402/v1/hamming-count","summary":"How many Hamming numbers are at most n. Through 1 there is 1, through 7 there are 6, through 10 there are 9, through 12 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hamming-next","url":"https://scorptech.it.com/api/x402/v1/hamming-next","summary":"The smallest Hamming number that is at least n. 7 gives 8, 10 stays 10, 11 gives 12, and 31 gives 32. n runs from 1 to 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/hamming-index","url":"https://scorptech.it.com/api/x402/v1/hamming-index","summary":"Where n sits in the Hamming sequence, counting from 1, or 0 if it is not Hamming. 1 is 1, 12 is 10, 36 is 20, and 7 is 0","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moran","url":"https://scorptech.it.com/api/x402/v1/moran","summary":"Whether n divided by its digit sum is a prime. 18, 21, and 27 are. 1, 10, 12, and 24 are not. n runs from 1 to 5000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moran-ratio","url":"https://scorptech.it.com/api/x402/v1/moran-ratio","summary":"n divided by its digit sum, when that division is exact. 18 gives 2, 21 gives 7, 12 gives 4, and 1 gives 1. A digit sum ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moran-count","url":"https://scorptech.it.com/api/x402/v1/moran-count","summary":"How many Moran numbers are at most n. Through 1 there are none, through 18 there is 1, through 21 there are 2, through 3","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moran-next","url":"https://scorptech.it.com/api/x402/v1/moran-next","summary":"The smallest Moran number that is at least n. 1 and 18 give 18, 19 gives 21, 22 gives 27, and 100 gives 111. n runs from","price":"$0.001","params":[]},{"method":"POST","path":"/v1/moran-sum","url":"https://scorptech.it.com/api/x402/v1/moran-sum","summary":"The sum of the Moran numbers that are at most n. Through 1 it is 0, through 18 it is 18, through 21 it is 39, through 30","price":"$0.001","params":[]},{"method":"POST","path":"/v1/smarandache","url":"https://scorptech.it.com/api/x402/v1/smarandache","summary":"The smallest positive k such that n divides k!. 1 gives 1, 6 gives 3, 8 gives 4, 9 gives 6, 12 gives 4, and 16 gives 6. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/smarandache-sum","url":"https://scorptech.it.com/api/x402/v1/smarandache-sum","summary":"The sum of S(k) from 1 through n. Through 1 it is 1, through 6 it is 18, through 10 it is 40, and through 12 it is 55. n","price":"$0.001","params":[]},{"method":"POST","path":"/v1/smarandache-quot","url":"https://scorptech.it.com/api/x402/v1/smarandache-quot","summary":"S(n)! divided by n. 1 gives 1, 6 gives 1, 8 gives 3, 9 gives 80, 10 gives 12, and 16 gives 45. n runs from 1 to 20.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/smarandache-gap","url":"https://scorptech.it.com/api/x402/v1/smarandache-gap","summary":"n minus S(n). 1 and 4 give 0, 6 gives 3, 8 gives 4, 10 gives 5, and 16 gives 10. n runs from 1 to 100000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/smarandache-least","url":"https://scorptech.it.com/api/x402/v1/smarandache-least","summary":"The smallest positive integer m with S(m) = k. k=1 gives 1, k=6 gives 9, k=8 gives 32, k=9 gives 27, and k=11 gives 11. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/alternating-factorial","url":"https://scorptech.it.com/api/x402/v1/alternating-factorial","summary":"The alternating factorial a(n) = n! - a(n-1), with a(1) = 1. a(3) is 5, a(4) is 19, a(5) is 101, a(6) is 619, and a(8) i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/alternating-sum","url":"https://scorptech.it.com/api/x402/v1/alternating-sum","summary":"The sum of the alternating factorials from 1 through n. Through 1 it is 1, through 3 it is 7, through 5 it is 127, throu","price":"$0.001","params":[]},{"method":"POST","path":"/v1/alternating-prime","url":"https://scorptech.it.com/api/x402/v1/alternating-prime","summary":"Whether the alternating factorial a(n) is prime. It is for 3, 4, 5, 6, 7, 8, and 10. It is not for 1, 2, 9, or 12. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/alternating-mod","url":"https://scorptech.it.com/api/x402/v1/alternating-mod","summary":"The alternating factorial a(n) modulo m. a(5) mod 10 is 1, a(6) mod 100 is 19, and a(4) mod 5 is 4. n runs from 1 to 18 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/alternating-next","url":"https://scorptech.it.com/api/x402/v1/alternating-next","summary":"The smallest m at least n, and at most 12, whose alternating factorial is prime. 1 gives 3, 3 stays 3, and 9 gives 10. I","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fibbinary","url":"https://scorptech.it.com/api/x402/v1/fibbinary","summary":"Whether the binary expansion of n has no two adjacent 1 bits. 0, 1, 2, 5, 10, and 21 do. 3, 6, and 7 do not. n runs from","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fibbinary-next","url":"https://scorptech.it.com/api/x402/v1/fibbinary-next","summary":"The smallest fibbinary number that is at least n. 0 stays 0, 3 gives 4, 6 and 7 give 8, 11 gives 16, and 21 stays 21. n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fibbinary-count","url":"https://scorptech.it.com/api/x402/v1/fibbinary-count","summary":"How many fibbinary numbers from 0 through n there are. Through 0 there is 1, through 2 there are 3, through 3 there are ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fibbinary-nth","url":"https://scorptech.it.com/api/x402/v1/fibbinary-nth","summary":"The nth fibbinary number, counting from 1, with 0 in position 1. The 1st is 0, the 4th is 4, the 8th is 10, the 10th is ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/fibbinary-index","url":"https://scorptech.it.com/api/x402/v1/fibbinary-index","summary":"The position of n among the fibbinary numbers, counting from 1, or 0 if n is not fibbinary. 0 is 1, 1 is 2, 5 is 5, 10 i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/arithmetic-number","url":"https://scorptech.it.com/api/x402/v1/arithmetic-number","summary":"Whether the average of the divisors of n is an integer. 1, 3, 6, 15, and 20 are. 2, 4, 8, and 12 are not. n runs from 1 ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/arithmetic-mean","url":"https://scorptech.it.com/api/x402/v1/arithmetic-mean","summary":"The average of the divisors of n, when that average is an integer. 1 gives 1, 3 gives 2, 6 gives 3, 7 gives 4, 15 gives ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/arithmetic-count","url":"https://scorptech.it.com/api/x402/v1/arithmetic-count","summary":"How many arithmetic numbers are at most n. Through 1 there is 1, through 2 there is 1, through 6 there are 4, through 15","price":"$0.001","params":[]},{"method":"POST","path":"/v1/arithmetic-next","url":"https://scorptech.it.com/api/x402/v1/arithmetic-next","summary":"The smallest arithmetic number that is at least n. 1 stays 1, 2 gives 3, 4 gives 5, 8 gives 11, and 13 stays 13. n runs ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/arithmetic-sum","url":"https://scorptech.it.com/api/x402/v1/arithmetic-sum","summary":"The sum of the arithmetic numbers that are at most n. Through 1 it is 1, through 6 it is 15, through 15 it is 75, and th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/apery","url":"https://scorptech.it.com/api/x402/v1/apery","summary":"The Apery number A(n), the sum of C(n,k)^2 * C(n+k,k)^2 for k from 0 to n. A(0) is 1, A(1) is 5, A(2) is 73, A(3) is 144","price":"$0.001","params":[]},{"method":"POST","path":"/v1/apery-sum","url":"https://scorptech.it.com/api/x402/v1/apery-sum","summary":"The sum of the Apery numbers A(0) through A(n). Through 0 it is 1, through 2 it is 79, and through 5 it is 853530. n run","price":"$0.001","params":[]},{"method":"POST","path":"/v1/apery-mod","url":"https://scorptech.it.com/api/x402/v1/apery-mod","summary":"A(n) modulo m. A(1) mod 2 is 1, A(5) mod 10 is 5, and A(8) mod 1000 is 425. n runs from 0 to 11 and m runs from 1 to 100","price":"$0.001","params":[]},{"method":"POST","path":"/v1/apery-term","url":"https://scorptech.it.com/api/x402/v1/apery-term","summary":"One summand of the Apery number, C(n,k)^2 * C(n+k,k)^2. The k=1 summand of A(2) is 36 and the k=0 summand is 1. A k past","price":"$0.001","params":[]},{"method":"POST","path":"/v1/apery-prime","url":"https://scorptech.it.com/api/x402/v1/apery-prime","summary":"Whether the Apery number A(n) is prime. A(1) and A(2) are prime. A(0), A(3), A(5), and A(8) are not. n runs from 0 to 11","price":"$0.001","params":[]},{"method":"POST","path":"/v1/franel","url":"https://scorptech.it.com/api/x402/v1/franel","summary":"The Franel number F(n), the sum of C(n,k)^3 for k from 0 to n. F(0) is 1, F(1) is 2, F(2) is 10, F(4) is 346, and F(5) i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/franel-sum","url":"https://scorptech.it.com/api/x402/v1/franel-sum","summary":"The sum of the Franel numbers F(0) through F(n). Through 0 it is 1, through 1 it is 3, through 4 it is 415, and through ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/franel-mod","url":"https://scorptech.it.com/api/x402/v1/franel-mod","summary":"F(n) modulo m. F(4) mod 7 is 3, F(5) mod 10 is 2, and F(8) mod 100 is 62. n runs from 0 to 16 and m runs from 1 to 10000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/franel-gcd","url":"https://scorptech.it.com/api/x402/v1/franel-gcd","summary":"The greatest common divisor of F(n) and F(n+1). n=0 gives 1, n=1 gives 2, and n=6 gives 16. n runs from 0 to 15.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/franel-central","url":"https://scorptech.it.com/api/x402/v1/franel-central","summary":"The middle summand of the Franel number, C(n, floor(n/2))^3. n=0 gives 1, n=2 gives 8, n=4 gives 216, n=5 gives 1000, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trinomial","url":"https://scorptech.it.com/api/x402/v1/trinomial","summary":"The central trinomial coefficient T(n), the coefficient of x^n in (1+x+x^2)^n. T(0) and T(1) are 1, T(2) is 3, T(4) is 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trinomial-sum","url":"https://scorptech.it.com/api/x402/v1/trinomial-sum","summary":"The sum of the central trinomial coefficients T(0) through T(n). Through 0 it is 1, through 1 it is 2, through 3 it is 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trinomial-mod","url":"https://scorptech.it.com/api/x402/v1/trinomial-mod","summary":"T(n) modulo m. T(5) mod 10 is 1, T(6) mod 100 is 41, and T(4) mod 5 is 4. n runs from 0 to 30 and m runs from 1 to 10000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trinomial-term","url":"https://scorptech.it.com/api/x402/v1/trinomial-term","summary":"The summand C(n,2k)*C(2k,k) of the central trinomial coefficient. For n=5 the summands are 1, 20, and 30. A k past n/2 i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/trinomial-prime","url":"https://scorptech.it.com/api/x402/v1/trinomial-prime","summary":"Whether T(n) is prime. T(2), T(3), and T(4) are prime. T(0), T(1), T(5), and T(6) are not. n runs from 0 to 30.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/domb","url":"https://scorptech.it.com/api/x402/v1/domb","summary":"The Domb number D(n), the sum of C(n,k)^2 * C(2k,k) * C(2n-2k,n-k) for k from 0 to n. D(0) is 1, D(1) is 4, D(2) is 28, ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/domb-sum","url":"https://scorptech.it.com/api/x402/v1/domb-sum","summary":"The sum of the Domb numbers D(0) through D(n). Through 0 it is 1, through 2 it is 33, through 4 it is 3005, and through ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/domb-mod","url":"https://scorptech.it.com/api/x402/v1/domb-mod","summary":"D(n) modulo m. D(5) mod 100 is 4, D(6) mod 1000 is 136, and D(1) mod 3 is 1. n runs from 0 to 12 and m runs from 1 to 10","price":"$0.001","params":[]},{"method":"POST","path":"/v1/domb-gcd","url":"https://scorptech.it.com/api/x402/v1/domb-gcd","summary":"The greatest common divisor of D(n) and D(n+1). n=0 gives 1, n=1 gives 4, and n=6 gives 64. n runs from 0 to 11.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/domb-term","url":"https://scorptech.it.com/api/x402/v1/domb-term","summary":"One summand of the Domb number, C(n,k)^2 * C(2k,k) * C(2n-2k,n-k). The k=1 summand of D(2) is 16 and the k=0 summand of ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/octahedral","url":"https://scorptech.it.com/api/x402/v1/octahedral","summary":"The nth octahedral number, n*(2n^2+1)/3. The first terms are 1, 6, 19, 44, and 85. n runs from 1 to 100000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/octahedral-sum","url":"https://scorptech.it.com/api/x402/v1/octahedral-sum","summary":"The sum of the first n octahedral numbers. Through 1 it is 1, through 3 it is 26, through 6 it is 301, and through 10 it","price":"$0.001","params":[]},{"method":"POST","path":"/v1/octahedral-index","url":"https://scorptech.it.com/api/x402/v1/octahedral-index","summary":"The n with O(n) equal to the input, or 0 when the input is not octahedral. 1 is place 1, 6 is place 2, 19 is place 3, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/octahedral-next","url":"https://scorptech.it.com/api/x402/v1/octahedral-next","summary":"The smallest octahedral number that is at least n. 1 stays 1, 2 gives 6, 7 gives 19, and 45 gives 85. n runs from 0 to 6","price":"$0.001","params":[]},{"method":"POST","path":"/v1/octahedral-mod","url":"https://scorptech.it.com/api/x402/v1/octahedral-mod","summary":"The nth octahedral number modulo m. O(4) mod 10 is 4, O(5) mod 7 is 1, and O(6) mod 10 is 6. n runs from 1 to 100000 and","price":"$0.001","params":[]},{"method":"POST","path":"/v1/egyptian","url":"https://scorptech.it.com/api/x402/v1/egyptian","summary":"The greedy Egyptian-fraction denominators of a/b, written comma-separated. 3/7 gives 3,11,231. 5/6 gives 2,3. 1/2 gives ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/egyptian-length","url":"https://scorptech.it.com/api/x402/v1/egyptian-length","summary":"How many unit fractions the greedy Egyptian expansion of a/b uses. 1/2 uses 1, 5/6 uses 2, 3/7 uses 3, and 8/11 uses 4. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/egyptian-first","url":"https://scorptech.it.com/api/x402/v1/egyptian-first","summary":"The first denominator in the greedy Egyptian expansion of a/b. 3/7 starts at 3, 5/6 starts at 2, and 2/3 starts at 2. Th","price":"$0.001","params":[]},{"method":"POST","path":"/v1/egyptian-last","url":"https://scorptech.it.com/api/x402/v1/egyptian-last","summary":"The last denominator in the greedy Egyptian expansion of a/b. 3/7 ends at 231, 8/11 ends at 4070, and 1/2 ends at 2. The","price":"$0.001","params":[]},{"method":"POST","path":"/v1/egyptian-sum","url":"https://scorptech.it.com/api/x402/v1/egyptian-sum","summary":"The sum of the denominators in the greedy Egyptian expansion of a/b. 3/7 sums to 245, 5/6 sums to 5, and 8/11 sums to 41","price":"$0.001","params":[]},{"method":"POST","path":"/v1/exp-sigma","url":"https://scorptech.it.com/api/x402/v1/exp-sigma","summary":"The sum of the exponential divisors of n. 1 gives 1, 8 gives 10, 12 gives 18, 16 gives 22, and 36 gives 72. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/exp-count","url":"https://scorptech.it.com/api/x402/v1/exp-count","summary":"How many exponential divisors n has. 1 and 6 have 1, 8 and 12 have 2, 16 has 3, and 36 has 4. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/exp-divisors","url":"https://scorptech.it.com/api/x402/v1/exp-divisors","summary":"The exponential divisors of n, comma-separated and ascending. 8 gives 2,8. 12 gives 6,12. 16 gives 2,4,16. 36 gives 6,12","price":"$0.001","params":[]},{"method":"POST","path":"/v1/exp-perfect","url":"https://scorptech.it.com/api/x402/v1/exp-perfect","summary":"Whether the exponential divisors of n sum to 2n. 36 and 180 do. 6, 12, 16, and 60 do not. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/exp-next","url":"https://scorptech.it.com/api/x402/v1/exp-next","summary":"The smallest exponential perfect number that is at least n. 1 gives 36, 37 gives 180, 180 stays 180, and 181 gives 252. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/infinitary-sigma","url":"https://scorptech.it.com/api/x402/v1/infinitary-sigma","summary":"The sum of the infinitary divisors of n. 6 gives 12, 8 gives 15, 16 gives 17, 36 gives 50, and 48 gives 68. n runs from ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/infinitary-count","url":"https://scorptech.it.com/api/x402/v1/infinitary-count","summary":"How many infinitary divisors n has. 6 and 8 and 12 have 4, 16 has 2, and 24 has 8. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/infinitary-perfect","url":"https://scorptech.it.com/api/x402/v1/infinitary-perfect","summary":"Whether the infinitary divisors of n sum to 2n. 6, 60, and 90 do. 16, 24, 36, and 48 do not. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/infinitary-abundant","url":"https://scorptech.it.com/api/x402/v1/infinitary-abundant","summary":"Whether the infinitary divisors of n sum to more than 2n. 24 and 30 do. 6, 16, 36, and 48 do not. n runs from 1 to 10000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/infinitary-deficient","url":"https://scorptech.it.com/api/x402/v1/infinitary-deficient","summary":"Whether the infinitary divisors of n sum to less than 2n. 1, 16, 36, and 48 do. 6, 24, and 60 do not. n runs from 1 to 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/duffinian","url":"https://scorptech.it.com/api/x402/v1/duffinian","summary":"Whether n is composite and coprime to its divisor sum. 4, 8, 9, 21, and 25 are. 1, 6, 7, and 15 are not. n runs from 1 t","price":"$0.001","params":[]},{"method":"POST","path":"/v1/duffinian-gcd","url":"https://scorptech.it.com/api/x402/v1/duffinian-gcd","summary":"The greatest common divisor of n and its divisor sum. 4 gives 1, 6 gives 6, 9 gives 1, and 15 gives 3. n runs from 1 to ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/duffinian-count","url":"https://scorptech.it.com/api/x402/v1/duffinian-count","summary":"How many Duffinian numbers are at most n. Through 1 there are 0, through 4 there is 1, through 9 there are 3, and throug","price":"$0.001","params":[]},{"method":"POST","path":"/v1/duffinian-next","url":"https://scorptech.it.com/api/x402/v1/duffinian-next","summary":"The smallest Duffinian number that is at least n. 1 gives 4, 5 gives 8, 10 gives 16, and 22 gives 25. n runs from 1 to 2","price":"$0.001","params":[]},{"method":"POST","path":"/v1/duffinian-sum","url":"https://scorptech.it.com/api/x402/v1/duffinian-sum","summary":"The sum of the Duffinian numbers that are at most n. Through 4 it is 4, through 9 it is 21, through 20 it is 37, and thr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sociable","url":"https://scorptech.it.com/api/x402/v1/sociable","summary":"Whether n lies on an aliquot cycle of length at least 3. 12496 does. 6, 220, and 25 do not. A chain past 30 steps or pas","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sociable-length","url":"https://scorptech.it.com/api/x402/v1/sociable-length","summary":"The length of the aliquot cycle containing n, or 0 when n is not on a cycle. 6 gives 1, 220 gives 2, 12496 gives 5, and ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sociable-min","url":"https://scorptech.it.com/api/x402/v1/sociable-min","summary":"The smallest member of the aliquot cycle containing n. 284 gives 220 and 14288 gives 12496. A number that is not on a cy","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sociable-sum","url":"https://scorptech.it.com/api/x402/v1/sociable-sum","summary":"The sum of the aliquot cycle containing n. 6 gives 6, 220 gives 504, and 12496 gives 71056. A number that is not on a cy","price":"$0.001","params":[]},{"method":"POST","path":"/v1/sociable-chain","url":"https://scorptech.it.com/api/x402/v1/sociable-chain","summary":"The aliquot cycle containing n, starting at n, comma-separated. 220 gives 220,284. 6 gives 6. 12496 gives 12496,14288,15","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pierpont","url":"https://scorptech.it.com/api/x402/v1/pierpont","summary":"Whether n is 2^a * 3^b + 1. 2, 4, 10, 19, and 145 are. 1, 8, and 11 are not. n runs from 1 to 999999999999999.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pierpont-prime","url":"https://scorptech.it.com/api/x402/v1/pierpont-prime","summary":"Whether n is a Pierpont prime, a prime of the form 2^a * 3^b + 1. 2, 7, 19, and 109 are. 4, 10, 11, and 145 are not. n r","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pierpont-count","url":"https://scorptech.it.com/api/x402/v1/pierpont-count","summary":"How many Pierpont primes are at most n. Through 1 there are 0, through 10 there are 4, through 20 there are 7, and throu","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pierpont-next","url":"https://scorptech.it.com/api/x402/v1/pierpont-next","summary":"The smallest Pierpont prime that is at least n. 1 gives 2, 8 gives 13, 20 gives 37, and 98 gives 109. n runs from 1 to 1","price":"$0.001","params":[]},{"method":"POST","path":"/v1/pierpont-powers","url":"https://scorptech.it.com/api/x402/v1/pierpont-powers","summary":"The pair a,b with n = 2^a * 3^b + 1. 2 gives 0,0. 10 gives 0,2. 19 gives 1,2. 145 gives 4,2. A number not of that form i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/parking","url":"https://scorptech.it.com/api/x402/v1/parking","summary":"Whether the preference list is a parking function. (1,1), (2,1), and (1,2,1) are. (2,2) and (1,3) are not. The list has ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/parking-spots","url":"https://scorptech.it.com/api/x402/v1/parking-spots","summary":"Where each car parks, comma-separated. (1,1) parks at 1,2. (2,1) parks at 2,1. (1,2,1) parks at 1,2,3. A list that is no","price":"$0.001","params":[]},{"method":"POST","path":"/v1/parking-shift","url":"https://scorptech.it.com/api/x402/v1/parking-shift","summary":"The total of parked spot minus preference. (1,1) shifts by 1. (2,1) shifts by 0. (1,2,1) shifts by 2. A list that is not","price":"$0.001","params":[]},{"method":"POST","path":"/v1/parking-moved","url":"https://scorptech.it.com/api/x402/v1/parking-moved","summary":"How many cars miss their preferred spot. (1,1) moves 1 car. (2,1) moves none. (1,2,1) moves 1. A list that is not a park","price":"$0.001","params":[]},{"method":"POST","path":"/v1/parking-witness","url":"https://scorptech.it.com/api/x402/v1/parking-witness","summary":"The first 1-based place where the sorted preferences exceed the place, or 0 when the list is a parking function. (1,1) g","price":"$0.001","params":[]},{"method":"POST","path":"/v1/blum","url":"https://scorptech.it.com/api/x402/v1/blum","summary":"Whether n is a product of two distinct primes that are both 3 mod 4. 21, 33, and 77 are. 9, 15, 25, 35, and 49 are not. ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/blum-factors","url":"https://scorptech.it.com/api/x402/v1/blum-factors","summary":"The two prime factors of a Blum integer, smaller first. 21 gives 3,7. 33 gives 3,11. 77 gives 7,11. A non-Blum integer i","price":"$0.001","params":[]},{"method":"POST","path":"/v1/blum-count","url":"https://scorptech.it.com/api/x402/v1/blum-count","summary":"How many Blum integers are at most n. Through 20 there are 0, through 21 there is 1, through 33 there are 2, and through","price":"$0.001","params":[]},{"method":"POST","path":"/v1/blum-next","url":"https://scorptech.it.com/api/x402/v1/blum-next","summary":"The smallest Blum integer that is at least n. 1 gives 21, 22 gives 33, 34 gives 57, and 78 gives 93. n runs from 1 to 10","price":"$0.001","params":[]},{"method":"POST","path":"/v1/blum-sum","url":"https://scorptech.it.com/api/x402/v1/blum-sum","summary":"The sum of the Blum integers that are at most n. Through 21 it is 21, through 33 it is 54, through 77 it is 257, and thr","price":"$0.001","params":[]},{"method":"POST","path":"/v1/carmichael-number","url":"https://scorptech.it.com/api/x402/v1/carmichael-number","summary":"Whether n is a Carmichael number: composite, squarefree, and p-1 divides n-1 for every prime p dividing n. 561, 1105, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/carmichael-factors","url":"https://scorptech.it.com/api/x402/v1/carmichael-factors","summary":"The prime factors of a Carmichael number, ascending and comma-separated. 561 gives 3,11,17. 1105 gives 5,13,17. 1729 giv","price":"$0.001","params":[]},{"method":"POST","path":"/v1/carmichael-count","url":"https://scorptech.it.com/api/x402/v1/carmichael-count","summary":"How many Carmichael numbers are at most n. Through 560 there are 0, through 561 there is 1, through 1729 there are 3, an","price":"$0.001","params":[]},{"method":"POST","path":"/v1/carmichael-next","url":"https://scorptech.it.com/api/x402/v1/carmichael-next","summary":"The smallest Carmichael number that is at least n. 1 gives 561, 562 gives 1105, 2000 gives 2465, and 2822 gives 6601. n ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/carmichael-sum","url":"https://scorptech.it.com/api/x402/v1/carmichael-sum","summary":"The sum of the Carmichael numbers that are at most n. Through 561 it is 561, through 1105 it is 1666, through 1729 it is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/biunitary-sigma","url":"https://scorptech.it.com/api/x402/v1/biunitary-sigma","summary":"The sum of the bi-unitary divisors of n. For a prime power p^e the exponents run from 0 through e, skipping the middle o","price":"$0.001","params":[]},{"method":"POST","path":"/v1/biunitary-count","url":"https://scorptech.it.com/api/x402/v1/biunitary-count","summary":"How many bi-unitary divisors n has. 1 has 1, 4 and 9 have 2, 8 has 4, and 16 has 4. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/biunitary-perfect","url":"https://scorptech.it.com/api/x402/v1/biunitary-perfect","summary":"Whether the bi-unitary divisors of n sum to 2n. 6, 60, and 90 do. 16, 24, 36, and 48 do not. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/biunitary-abundant","url":"https://scorptech.it.com/api/x402/v1/biunitary-abundant","summary":"Whether the bi-unitary divisors of n sum to more than 2n. 24 and 48 do. 6, 16, and 36 do not. n runs from 1 to 1000000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/biunitary-deficient","url":"https://scorptech.it.com/api/x402/v1/biunitary-deficient","summary":"Whether the bi-unitary divisors of n sum to less than 2n. 1, 16, and 36 do. 6, 24, and 48 do not. n runs from 1 to 10000","price":"$0.001","params":[]},{"method":"POST","path":"/v1/higgs","url":"https://scorptech.it.com/api/x402/v1/higgs","summary":"Whether n is a Higgs prime for exponent 2. 2, 13, and 19 are. 4, 9, and 17 are not. n runs from 1 to 2000.","price":"$0.001","params":[]},{"method":"POST","path":"/v1/higgs-count","url":"https://scorptech.it.com/api/x402/v1/higgs-count","summary":"How many Higgs primes for exponent 2 are at most n. Through 1 there are 0, through 16 there are 6, through 19 there are ","price":"$0.001","params":[]},{"method":"POST","path":"/v1/higgs-next","url":"https://scorptech.it.com/api/x402/v1/higgs-next","summary":"The smallest Higgs prime for exponent 2 that is at least n. 1 gives 2, 14 gives 19, 17 gives 19, and 73 gives 79. n runs","price":"$0.001","params":[]},{"method":"POST","path":"/v1/higgs-index","url":"https://scorptech.it.com/api/x402/v1/higgs-index","summary":"The 1-based place of n in the Higgs primes for exponent 2, or 0 when n is absent. 2 is place 1, 19 is place 7, and 17 is","price":"$0.001","params":[]},{"method":"POST","path":"/v1/higgs-gap","url":"https://scorptech.it.com/api/x402/v1/higgs-gap","summary":"The distance from a Higgs prime to the next one. 2 gives 1, 7 gives 4, 13 gives 6, and 71 gives 8. A number that is not ","price":"$0.001","params":[]}]}