{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.practical-number","code":"thing-q577211","url":"https://ver.cy/models/thing/q577211/","name":"practical number","alternateNames":[],"kind":"thing","status":"published","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"A positive integer n such that every smaller positive integer can be written as a sum of distinct divisors of n, the sequence beginning 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, named by Srinivasan in 1948, characterised by a condition on prime factorisation by Stewart and Sierpinski, and including all powers of two and all perfect numbers; practical numbers behave in some ways like primes, and Goldbach-type results and density estimates have been proved for them.","purpose":"Let an agent define practical numbers, relay their characterisation, properties and known results from number theory references, and distinguish them from related classes such as perfect and abundant numbers.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Subset sum property of divisors","Stewart-Sierpinski characterisation","Prime-like distribution","All even except 1"],"structure":{"bundles":[{"id":"define","name":"Define","description":"Definition.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a practical number, and how is it checked?","kind":"definition"},{"text":"Is the question about practical, perfect or abundant numbers?","kind":"boundary"}]}]},{"id":"characterisation","name":"Characterisation","description":"Characterisation.","findings":[{"id":"characterisation-finding","name":"Characterisation","description":"Characterisation.","questions":[{"text":"What is the Stewart-Sierpinski characterisation in terms of prime factorisation?","kind":"definition"},{"text":"Which entry fits prime factorisation?","kind":"action"}]}]}]},{"id":"results","name":"Results","description":"Results.","layers":[{"id":"properties","name":"Properties","description":"Properties.","findings":[{"id":"properties-finding","name":"Properties","description":"Properties.","questions":[{"text":"What properties do practical numbers have, including closure under products?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"primes","name":"Primes","description":"Analogies with primes.","findings":[{"id":"primes-finding","name":"Primes","description":"Primes.","questions":[{"text":"What Goldbach-type and density results hold for practical numbers, as proved by Melfi, Weingartner and others?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"apply","name":"Apply","description":"Applications.","layers":[{"id":"egyptian","name":"Egyptian","description":"Egyptian fractions.","findings":[{"id":"egyptian-finding","name":"Egyptian","description":"Egyptian.","questions":[{"text":"How are practical numbers used in Egyptian fraction representations?","kind":"provenance"},{"text":"Which entry fits Egyptian fraction?","kind":"action"}]}]},{"id":"computation","name":"Computation","description":"Computation.","findings":[{"id":"computation-finding","name":"Computation","description":"Computation.","questions":[{"text":"How are practical numbers computed and enumerated?","kind":"provenance"},{"text":"Which entry fits computational number theory?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Srinivasan, Stewart and Sierpinski introduce and characterise practical numbers?","kind":"provenance"},{"text":"Which entry fits the history of number theory?","kind":"action"}]}]},{"id":"related","name":"Related","description":"Related classes.","findings":[{"id":"related-finding","name":"Related","description":"Related.","questions":[{"text":"How do practical numbers relate to perfect, abundant and highly composite numbers?","kind":"provenance"},{"text":"Which entry fits perfect number?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Test whether an integer is practical and list its divisors as a witness.","Explain properties of practical numbers and their link to Egyptian fractions."],"mustNot":["Present a conjectured property as proven.","Confuse practical numbers with similar sequences such as highly composite numbers.","Credit the naming to someone other than Srinivasan."],"requiresHuman":["Publishing new results on practical numbers as established."]},"ethics":{"considerations":["Accurate credit and proof standards keep mathematics trustworthy.","Students rely on correct definitions."],"affectedParties":["Mathematicians","Students"]},"owners":{"steward":"The public: held in common as mathematical knowledge.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"in registry terms"},{"target":"prime factorization","type":"related","note":"via the Stewart-Sierpinski condition"},{"target":"perfect number","type":"related","note":"as a subclass"},{"target":"OEIS","type":"related","note":"as A005153"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.practical-number","Wikidata: Q577211 (https://www.wikidata.org/wiki/Q577211)","OEIS: A005153"]},"properties":{"applicability":"not-applicable","items":["smallest: 1 number","smallest odd practical number: 1 number - the only odd one"]},"recognition":{"applicability":"optional","items":["Integers whose divisors sum to all smaller integers","Sequence OEIS A005153","Perfect numbers equal their aliquot sum; abundant numbers exceed it","The sequence 1, 2, 4, 6, 8, 12, 16, 18, 20, 24, 28, 30, ..."]},"capabilities":{"applicability":"required","items":["define and characterise","relay properties and results","relay analogies with primes","distinguish related classes"]},"hazards":{"applicability":"required","items":["Confusing with perfect or abundant numbers","Misstating the characterisation","Computation errors"]},"interfaces":{"applicability":"required","items":["No regulation; mathematical conventions"]},"context":{"applicability":"required","items":["Not applicable; a class of integers.","practical numbers in general","powers of two and perfect numbers as practical numbers","primitive practical numbers","practical numbers in Egyptian fraction problems"]}},"sources":[{"title":"Wikidata item Q577211: practical number","url":"https://www.wikidata.org/wiki/Q577211","note":"identity and sense of the item"},{"title":"Wikipedia: Practical number","url":"https://en.wikipedia.org/wiki/Practical_number","note":"general description of the item"}],"openQuestions":["How should the density and Goldbach-type results be linked?","How should OEIS and literature be linked?","How should primitive practical numbers be recorded?"],"resources":{"spec":"/models/things/publications/thing-q577211/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q577211/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T02:17:19Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:52:01+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}