{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.refactorable-number","code":"thing-q2063121","url":"https://ver.cy/models/thing/q2063121/","name":"refactorable 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 that is divisible by the number of its divisors, also called a tau number, such as 1, 2, 8, 9, 12, 18, 24, 36, 40 and 56, so that 12 with six divisors is refactorable because 6 divides 12; refactorable numbers have natural density zero, can occur in consecutive pairs such as 8 and 9, and were studied by Kennedy and Cooper and by Zelinsky.","purpose":"Let an agent explain refactorable numbers, relay their definition, examples and known properties from number theory references, and distinguish them from related divisor-based number classes.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["The divisor count divides the number.","Refactorable numbers have natural density zero.","Consecutive refactorable numbers include 8 and 9.","The definition uses the divisor-counting tau function."],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What a refactorable number is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a refactorable number, and why is 12 refactorable while 10 is not?","kind":"definition"},{"text":"Is the question about refactorable numbers or another divisor-based class?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the first refactorable numbers, and which perfect squares and prime powers qualify?","kind":"measurement"},{"text":"Which entry fits the specific number?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Results.","layers":[{"id":"results","name":"Results","description":"Known results.","findings":[{"id":"results-finding","name":"Results","description":"Results.","questions":[{"text":"What is proved about density and consecutive refactorable numbers, and by whom?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"open","name":"Open","description":"Open questions.","findings":[{"id":"open-finding","name":"Open","description":"Open.","questions":[{"text":"What questions about refactorable numbers remain open?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"compare","name":"Compare","description":"Related classes.","layers":[{"id":"divisor","name":"Divisor","description":"Divisor function.","findings":[{"id":"divisor-finding","name":"Divisor","description":"Divisor.","questions":[{"text":"How does the divisor counting function tau behave, and how does it determine refactorability?","kind":"provenance"},{"text":"Which entry fits divisor function?","kind":"action"}]}]},{"id":"classes","name":"Classes","description":"Other classes.","findings":[{"id":"classes-finding","name":"Classes","description":"Classes.","questions":[{"text":"How do refactorable numbers relate to highly composite, perfect and harmonic divisor numbers?","kind":"provenance"},{"text":"Which entry fits highly composite number?","kind":"action"}]}]}]},{"id":"context","name":"Context","description":"History and computation.","layers":[{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How were refactorable numbers introduced by Kennedy and Cooper and named tau numbers by Colton?","kind":"provenance"},{"text":"Which entry fits the history of number theory?","kind":"action"}]}]},{"id":"computation","name":"Computation","description":"Computation.","findings":[{"id":"computation-finding","name":"Computation","description":"Computation.","questions":[{"text":"How can refactorable numbers be computed programmatically?","kind":"action"},{"text":"Which entry fits integer sequence?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Explain refactorable numbers and test given integers.","Count divisors correctly and show the working.","Cite known results with their authors."],"mustNot":["Miscount divisors or present wrong examples.","Overstate open results as proved.","Confuse tau numbers with other divisor classes.","Misattribute results."],"requiresHuman":["Publishing a new claimed theorem about refactorable numbers."]},"ethics":{"considerations":["Correct credit for results matters to mathematicians.","Errors in teaching material mislead students."],"affectedParties":["Students","Researchers","Readers"]},"owners":{"steward":"Nobody owns the concept; the mathematical community maintains it.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"in registry terms"},{"target":"divisor function","type":"related","note":"tau(n)"},{"target":"OEIS","type":"related","note":"as A033950"},{"target":"perfect number","type":"related","note":"defined by divisor sums"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.refactorable-number","Wikidata: Q2063121 (https://www.wikidata.org/wiki/Q2063121)","OEIS: A033950 refactorable numbers"]},"properties":{"applicability":"not-applicable","items":["first terms: 1, 2, 8, 9, 12, 18, 24, 36, 40, 56 sequence - OEIS A033950","natural density: 0 value - proved by Kennedy and Cooper","consecutive refactorable numbers: only 1 and 2 result - Zelinsky"]},"recognition":{"applicability":"optional","items":["Integers divisible by their number of divisors","Tau numbers 1, 2, 8, 9, 12, 18, 24, 36, 40, 56","Perfect numbers use divisor sums; highly composite numbers maximise divisor counts","Not a visible object; integers with a divisibility property."]},"capabilities":{"applicability":"required","items":["explain the definition","relay examples and properties","describe results","distinguish related classes"]},"hazards":{"applicability":"required","items":["Confusing tau numbers with other divisor classes","Miscounting divisors","Overstating open results"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical concept.","refactorable numbers","refactorable numbers that are perfect squares","refactorable primes powers such as 8 and 9","related classes such as highly composite and tau-rich numbers"]}},"sources":[{"title":"Wikidata item Q2063121: refactorable number","url":"https://www.wikidata.org/wiki/Q2063121","note":"identity and sense of the item"},{"title":"Wikipedia: Refactorable number","url":"https://en.wikipedia.org/wiki/Refactorable_number","note":"general description of the item"}],"openQuestions":["Should the divisor function be the primary linked entry?","How should number theory references be linked?","How should open problems such as Colton conjectures be recorded?"],"resources":{"spec":"/models/things/publications/thing-q2063121/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q2063121/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T03:52:47Z","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 2026-10-07T10:10:28+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}}