{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.highly-abundant-number","code":"thing-q3205712","url":"https://ver.cy/models/thing/q3205712/","name":"highly abundant 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 whose sum of divisors, including n itself, is greater than the sum of divisors of every smaller positive integer, the sequence beginning 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60; highly abundant numbers were introduced by Pillai in 1943 and studied by Alaoglu and Erdos, are related to superabundant and highly composite numbers, and are catalogued in the OEIS.","purpose":"Let an agent define highly abundant numbers, relay their sequence and known properties from number theory references, and distinguish them from abundant, superabundant and highly composite numbers.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Record divisor sum","Introduced by Pillai","Related to superabundant numbers","Not all even"],"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 highly abundant number, and how is it checked?","kind":"definition"},{"text":"Is the question about highly abundant, abundant, superabundant or highly composite numbers?","kind":"boundary"}]}]},{"id":"sequence","name":"Sequence","description":"Sequence.","findings":[{"id":"sequence-finding","name":"Sequence","description":"Sequence.","questions":[{"text":"What are the first highly abundant numbers, and where is the sequence catalogued?","kind":"measurement"},{"text":"Which entry fits divisor function?","kind":"action"}]}]}]},{"id":"results","name":"Results","description":"Results.","layers":[{"id":"properties","name":"Properties","description":"Known properties.","findings":[{"id":"properties-finding","name":"Properties","description":"Properties.","questions":[{"text":"What did Alaoglu and Erdos prove about highly abundant numbers?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"relations","name":"Relations","description":"Relations to other classes.","findings":[{"id":"relations-finding","name":"Relations","description":"Relations.","questions":[{"text":"How do highly abundant numbers relate to superabundant and highly composite numbers?","kind":"provenance"},{"text":"Which entry fits superabundant number?","kind":"action"}]}]}]},{"id":"compute","name":"Compute","description":"Computation.","layers":[{"id":"algorithms","name":"Algorithms","description":"Algorithms.","findings":[{"id":"algorithms-finding","name":"Algorithms","description":"Algorithms.","questions":[{"text":"How are highly abundant numbers computed?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]},{"id":"open","name":"Open","description":"Open questions.","findings":[{"id":"open-finding","name":"Open","description":"Open.","questions":[{"text":"What open questions exist about odd members and asymptotics?","kind":"provenance"},{"text":"Which entry fits open problems in 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 Pillai, Alaoglu and Erdos introduce and study these numbers?","kind":"provenance"},{"text":"Which entry fits the history of number theory?","kind":"action"}]}]},{"id":"related","name":"Related","description":"Record-setting classes.","findings":[{"id":"related-finding","name":"Related","description":"Related.","questions":[{"text":"What other record-setting integer classes exist, from Ramanujan highly composite numbers onward?","kind":"provenance"},{"text":"Which entry fits highly composite number?","kind":"action"}]}]}]}]},"agentConduct":{"may":["State the defining property of highly abundant numbers and the start of the sequence.","Compute divisor sums and test the property within stated limits."],"mustNot":["Confuse highly abundant numbers with abundant, superabundant or highly composite numbers.","Attribute the concept to the wrong mathematician.","Publish computed terms without the search bound and method.","Present an unverified conjecture about the sequence as proven."],"requiresHuman":["Publishing a claimed new result about the sequence."]},"ethics":{"considerations":["Similar-sounding number classes are easily mixed up, which misleads learners.","Attribution of mathematical ideas matters to the record."],"affectedParties":["Mathematicians","Students","Sequence database editors"]},"owners":{"steward":"Nobody owns a mathematical fact; sequence databases record it by convention.","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":"sigma(n)"},{"target":"superabundant number","type":"related","note":"a similar record class"},{"target":"OEIS","type":"related","note":"as A002093"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.highly-abundant-number","Wikidata: Q3205712 (https://www.wikidata.org/wiki/Q3205712)","OEIS: A002093"]},"properties":{"applicability":"not-applicable","items":["smallest: 1 number"]},"recognition":{"applicability":"optional","items":["Record-setting divisor sums","Sequence OEIS A002093","Abundant numbers exceed twice themselves; superabundant numbers set records for sigma(n)/n","The sequence 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, ..."]},"capabilities":{"applicability":"required","items":["define and list","relay known properties","relate to similar classes","distinguish from related concepts"]},"hazards":{"applicability":"required","items":["Confusing with abundant or highly composite numbers","Assuming all superabundant numbers are highly abundant or vice versa","Computation errors"]},"interfaces":{"applicability":"required","items":["No regulation; mathematical conventions"]},"context":{"applicability":"required","items":["Not applicable; a class of integers.","highly abundant numbers in general","superabundant numbers as a subsequence in many cases","highly composite numbers as a related record class","colossally abundant numbers"]}},"sources":[{"title":"Wikidata item Q3205712: highly abundant number","url":"https://www.wikidata.org/wiki/Q3205712","note":"identity and sense of the item"},{"title":"Wikipedia: Highly abundant number","url":"https://en.wikipedia.org/wiki/Highly_abundant_number","note":"general description of the item"}],"openQuestions":["What is the relation between highly abundant and superabundant numbers beyond known results?","How should OEIS and literature be linked?","Are there infinitely many odd highly abundant numbers?"],"resources":{"spec":"/models/things/publications/thing-q3205712/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q3205712/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-13T01:10:31Z","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 2026-10-06T19:51:58+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}}