{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.friendly-number","code":"thing-q1707307","url":"https://ver.cy/models/thing/q1707307/","name":"friendly 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 shares its abundancy index, the ratio of the sum of its divisors to itself, with at least one other integer, so that the two form a friendly pair or club, as 6 and 28 with abundancy 2, while an integer with no such partner is solitary, as are all primes and prime powers; whether numbers such as 10 are friendly or solitary is unknown, and the concept is distinct from amicable numbers.","purpose":"Let an agent explain friendly numbers and abundancy, relay known results and open problems from number theory references, and distinguish friendly numbers from amicable, perfect and sociable numbers.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Abundancy index","Pairs and clubs","Solitary numbers","Open cases"],"structure":{"bundles":[{"id":"understand","name":"Understand","description":"What a friendly number is.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a friendly number, what is abundancy, and what is a solitary number?","kind":"definition"},{"text":"Is the question about friendly numbers, amicable numbers or perfect numbers?","kind":"boundary"}]}]},{"id":"examples","name":"Examples","description":"Examples.","findings":[{"id":"examples-finding","name":"Examples","description":"Examples.","questions":[{"text":"What are examples of friendly pairs and clubs, and which small numbers are known to be solitary?","kind":"measurement"},{"text":"Which entry fits the specific number class?","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 solitary numbers, such as those coprime to their divisor sum?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]},{"id":"open","name":"Open","description":"Open problems.","findings":[{"id":"open-finding","name":"Open","description":"Open.","questions":[{"text":"Why is the status of 10 unknown, and what would resolve it?","kind":"provenance"},{"text":"Which sources are cited?","kind":"provenance"}]}]}]},{"id":"compare","name":"Compare","description":"Related concepts.","layers":[{"id":"amicable","name":"Amicable","description":"Amicable and sociable numbers.","findings":[{"id":"amicable-finding","name":"Amicable","description":"Amicable.","questions":[{"text":"How do amicable, sociable and perfect numbers relate to friendly numbers?","kind":"provenance"},{"text":"Which entry fits amicable number?","kind":"action"}]}]},{"id":"abundancy","name":"Abundancy","description":"Abundancy index.","findings":[{"id":"abundancy-finding","name":"Abundancy","description":"Abundancy.","questions":[{"text":"What properties does the abundancy index have, and how does it relate to abundant and deficient numbers?","kind":"provenance"},{"text":"Which entry fits abundant 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 did Anderson and Hickerson introduce friendly numbers, and how has the study developed?","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 friendly numbers be searched for computationally?","kind":"action"},{"text":"Which entry fits computational number theory?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Explain friendly numbers and the abundancy index with examples.","Check whether given numbers form a friendly pair.","State which numbers, such as 10, are unresolved."],"mustNot":["Claim that unresolved numbers are solitary or friendly without proof.","Confuse friendly numbers with amicable numbers.","Present computational searches as proofs.","Invent results or credit them wrongly."],"requiresHuman":["Publishing a claimed resolution of an open case."]},"ethics":{"considerations":["Mathematical honesty requires separating what is proved from what is believed.","Credit for results matters to researchers."],"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":"abundancy index","type":"related","note":"sigma(n)/n"},{"target":"amicable number","type":"related","note":"a different divisor relation"},{"target":"perfect number","type":"related","note":"as friendly numbers of abundancy 2"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.friendly-number","Wikidata: Q1707307 (https://www.wikidata.org/wiki/Q1707307)","OEIS: A074902 friendly numbers","OEIS: A095739 known solitary numbers"]},"properties":{"applicability":"not-applicable","items":["abundancy of 6 and 28: 2 value - all perfect numbers are friendly","smallest unresolved number: 10 value - unknown whether friendly or solitary","concept introduced: 1977 year - Anderson and Hickerson"]},"recognition":{"applicability":"optional","items":["Integers sharing an abundancy index","Friendly pairs and clubs; solitary numbers","Amicable numbers have divisor sums equal to each other; perfect numbers have abundancy exactly 2","Not a visible object; integers grouped by abundancy."]},"capabilities":{"applicability":"required","items":["explain the definition","relay known results","describe open problems","distinguish from related numbers"]},"hazards":{"applicability":"required","items":["Confusing friendly with amicable numbers","Assuming unresolved numbers are solitary","Misreading abundancy as a ratio of sums"]},"interfaces":{"applicability":"required","items":["No regulation; standard mathematical definitions"]},"context":{"applicability":"required","items":["Not applicable; a mathematical concept.","friendly pairs","friendly clubs with three or more members","solitary numbers including primes and prime powers","perfect numbers as a friendly club with abundancy 2","unresolved numbers such as 10, 14, 15, 20"]}},"sources":[{"title":"Wikidata item Q1707307: friendly number","url":"https://www.wikidata.org/wiki/Q1707307","note":"identity and sense of the item"},{"title":"Wikipedia: Friendly number","url":"https://en.wikipedia.org/wiki/Friendly_number","note":"general description of the item"}],"openQuestions":["Should abundancy index be a separate entry?","How should number theory references be linked?","How should the status of unresolved numbers be tracked?"],"resources":{"spec":"/models/things/publications/thing-q1707307/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q1707307/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 2026-10-06T19:51:55+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}}