{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.harmonic-divisor-number","code":"thing-q60679","url":"https://ver.cy/models/thing/q60679/","name":"harmonic divisor 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 whose divisors have an integer harmonic mean, also called an Ore number after Oystein Ore who introduced them in 1948; the first are 1, 6, 28, 140, 270 and 496, every perfect number is a harmonic divisor number, and it is conjectured that no odd harmonic divisor number greater than 1 exists.","purpose":"Let an agent define harmonic divisor numbers precisely, test a given integer, relate them to perfect numbers and open conjectures, and point to sequence references.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Defined by the harmonic mean of divisors","Contains all perfect numbers","Conjectured to have no odd members beyond 1","Named after Ore"],"structure":{"bundles":[{"id":"define","name":"Define","description":"The definition.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is the harmonic mean of the divisors, and when is it an integer?","kind":"definition"},{"text":"Is the question about harmonic divisor numbers or a related class?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are the harmonic divisor numbers below a given bound?","kind":"measurement"},{"text":"Which entry fits integer sequences?","kind":"action"}]}]}]},{"id":"theory","name":"Theory","description":"Properties.","layers":[{"id":"perfect","name":"Perfect","description":"Perfect numbers.","findings":[{"id":"perfect-finding","name":"Perfect","description":"Perfect.","questions":[{"text":"Why is every perfect number a harmonic divisor number?","kind":"provenance"},{"text":"Which entry fits perfect numbers?","kind":"action"}]}]},{"id":"conjectures","name":"Conjectures","description":"Open problems.","findings":[{"id":"conjectures-finding","name":"Conjectures","description":"Conjectures.","questions":[{"text":"What is known and conjectured about odd harmonic divisor numbers and their density?","kind":"provenance"},{"text":"What is proven versus conjectured?","kind":"boundary"}]}]}]},{"id":"compute","name":"Compute","description":"Computation.","layers":[{"id":"test","name":"Test","description":"Testing.","findings":[{"id":"test-finding","name":"Test","description":"Test.","questions":[{"text":"How is an integer tested using the number and sum of divisors?","kind":"action"},{"text":"What is the result for the number in question?","kind":"measurement"}]}]},{"id":"search","name":"Search","description":"Searches.","findings":[{"id":"search-finding","name":"Search","description":"Search.","questions":[{"text":"How far have harmonic divisor numbers been enumerated, with attribution?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]}]},{"id":"context","name":"Context","description":"History and teaching.","layers":[{"id":"history","name":"History","description":"Ore.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Ore introduce these numbers, and how did later work develop them?","kind":"provenance"},{"text":"Which entry fits Oystein Ore?","kind":"action"}]}]},{"id":"teach","name":"Teach","description":"Teaching.","findings":[{"id":"teach-finding","name":"Teach","description":"Teaching.","questions":[{"text":"How are harmonic divisor numbers used in teaching?","kind":"action"},{"text":"Which misconceptions arise?","kind":"provenance"}]}]}]}]},"agentConduct":{"may":["Define harmonic divisor numbers and test whether a given integer is one.","Relate them to perfect numbers and cite sequence references.","Explain the conjecture about odd harmonic divisor numbers as still open."],"mustNot":["Present the conjecture that no odd harmonic divisor number exceeds 1 as proven.","Report a number as a harmonic divisor number without checking its harmonic mean exactly.","Confuse harmonic divisor numbers with arithmetic or refactorable numbers.","Attribute results to Oystein Ore or other mathematicians that they did not publish."],"requiresHuman":["Announcing a claimed proof or counterexample as a new result."]},"ethics":{"considerations":["Accurate attribution respects the mathematicians who did the work.","Overstated claims in number theory mislead students and waste other people's checking time."],"affectedParties":["Students and learners","Mathematicians","Readers of references"]},"owners":{"steward":"Nobody owns the concept; the mathematical community and sequence databases maintain the record.","roles":[],"masterSystems":[]},"relations":[{"target":"Q16317911","type":"parent","note":"registry parent class"},{"target":"positive integer","type":"related","note":"a property of integers"},{"target":"perfect number","type":"related","note":"every perfect number qualifies"},{"target":"On-Line Encyclopedia of Integer Sequences","type":"related","note":"sequence A001599"},{"target":"harmonic mean","type":"related","note":"the defining average"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.harmonic-divisor-number","Wikidata: Q60679 (https://www.wikidata.org/wiki/Q60679)","OEIS: A001599 harmonic or Ore numbers"]},"properties":{"applicability":"not-applicable","items":["first terms: 1, 6, 28, 140, 270, 496, 672 integers"]},"recognition":{"applicability":"optional","items":["Divisors have integer harmonic mean","Examples 1, 6, 28, 140, 270, 496","Perfect numbers are a subset; abundant numbers are a different notion","Not visible; a property of integers."]},"capabilities":{"applicability":"required","items":["test whether an integer is a harmonic divisor number","list small examples","explain the link to perfect numbers","state open problems"]},"hazards":{"applicability":"required","items":["Confusing with arithmetic or refactorable numbers","Presenting the odd conjecture as proven","Errors in computing the harmonic mean"]},"interfaces":{"applicability":"required","items":["No regulation; definitions follow the literature and the OEIS"]},"context":{"applicability":"required","items":["Not for anything; a curiosity in number theory.","harmonic divisor numbers","perfect numbers as a subset","related notions such as refactorable and arithmetic numbers"]}},"sources":[{"title":"Wikidata item Q60679: harmonic divisor number","url":"https://www.wikidata.org/wiki/Q60679","note":"identity and sense of the item"},{"title":"Wikipedia: Harmonic divisor number","url":"https://en.wikipedia.org/wiki/Harmonic_divisor_number","note":"general description of the item"}],"openQuestions":["Should perfect numbers be linked as the primary related entry?","How should OEIS entries be linked?","How should the odd conjecture be recorded?"],"resources":{"spec":"/models/things/publications/thing-q60679/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q60679/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T21:03:38Z","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}}