{
  "vercy": "1.0-draft",
  "publication": {
    "status": "research-draft",
    "adjudicationStatus": "unreviewed",
    "publishableCanonical": false,
    "generatedAt": "2026-09-13T03:52:47Z",
    "providers": [
      "Claude"
    ],
    "breadth": "written by Claude from model knowledge without web access - no source was read, every claim is a lead to verify",
    "pass": 2,
    "wave": 3,
    "engine": "claude",
    "enrichment": {
      "pass": 3,
      "engine": "claude",
      "producedAt": "2026-10-06T10:00:00Z",
      "ingestedAt": "2026-10-06T07:09:00+00:00",
      "fields": [
        "agent_conduct",
        "ethics",
        "owners"
      ]
    },
    "gate": {
      "passed": true,
      "version": "1.0",
      "at": "2026-10-06T19:51:55+00:00",
      "evidence": [
        "completeness 1.00",
        "sense: Wikidata description",
        "2 live sources",
        "review by codex: pass"
      ]
    }
  },
  "metaModel": {
    "id": "THING-Q1707307",
    "registryId": "vr.tr.friendly-number",
    "name": "friendly number",
    "version": "0.2.0-wave.3",
    "entryKind": "thing",
    "family": "Thing Registry",
    "domain": [
      "XCT.QTY"
    ],
    "status": "research-draft"
  },
  "canonicalUrl": "https://ver.cy/models/thing/q1707307/",
  "model": {
    "registry_id": "vr.tr.friendly-number",
    "name": "friendly number",
    "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.",
    "definition": "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.",
    "what_it_is_for": "Not applicable; a mathematical concept.",
    "affordances": [
      "explain the definition",
      "relay known results",
      "describe open problems",
      "distinguish from related numbers"
    ],
    "distinguishing_features": [
      "Abundancy index",
      "Pairs and clubs",
      "Solitary numbers",
      "Open cases"
    ],
    "appearance": "Not a visible object; integers grouped by abundancy.",
    "visual_identification": [
      "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"
    ],
    "physical_properties": [
      {
        "quantity": "abundancy of 6 and 28",
        "typical_range": "2",
        "unit": "value",
        "note": "all perfect numbers are friendly"
      },
      {
        "quantity": "smallest unresolved number",
        "typical_range": "10",
        "unit": "value",
        "note": "unknown whether friendly or solitary"
      },
      {
        "quantity": "concept introduced",
        "typical_range": "1977",
        "unit": "year",
        "note": "Anderson and Hickerson"
      }
    ],
    "families_and_kinds": [
      "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"
    ],
    "related_models": [
      {
        "relation": "is a kind of",
        "target": "positive integer",
        "why": "in registry terms"
      },
      {
        "relation": "is defined by",
        "target": "abundancy index",
        "why": "sigma(n)/n"
      },
      {
        "relation": "is contrasted with",
        "target": "amicable number",
        "why": "a different divisor relation"
      },
      {
        "relation": "includes",
        "target": "perfect number",
        "why": "as friendly numbers of abundancy 2"
      }
    ],
    "identifiers": [
      {
        "scheme": "OEIS",
        "value_or_pattern": "A074902",
        "note": "friendly numbers"
      },
      {
        "scheme": "OEIS",
        "value_or_pattern": "A095739",
        "note": "known solitary numbers"
      }
    ],
    "standards_and_regulation": [
      "No regulation; standard mathematical definitions"
    ],
    "failure_modes_and_hazards": [
      "Confusing friendly with amicable numbers",
      "Assuming unresolved numbers are solitary",
      "Misreading abundancy as a ratio of sums"
    ],
    "in_scope": [],
    "out_of_scope": [],
    "characteristics": [],
    "agent_conduct": {
      "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."
      ],
      "must_not": [
        "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."
      ],
      "requires_human": [
        "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."
      ],
      "affected_parties": [
        "Students",
        "Researchers",
        "Readers"
      ]
    },
    "owners": {
      "steward": "Nobody owns the concept; the mathematical community maintains it.",
      "master_systems": []
    }
  },
  "sources": [
    {
      "id": "wikidata",
      "title": "Wikidata item Q1707307: friendly number",
      "url": "https://www.wikidata.org/wiki/Q1707307",
      "url_status": "live",
      "what_it_supports": "identity and sense of the item",
      "checked_by": "wikidata-api"
    },
    {
      "id": "wikipedia-en",
      "title": "Wikipedia: Friendly number",
      "url": "https://en.wikipedia.org/wiki/Friendly_number",
      "url_status": "live",
      "what_it_supports": "general description of the item",
      "checked_by": "wikidata-api sitelink"
    }
  ],
  "structure": {
    "bundles": [
      {
        "id": "understand",
        "name": "Understand",
        "description": "What a friendly number is.",
        "rationale": "Mathematics.",
        "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.",
        "rationale": "Mathematics.",
        "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.",
        "rationale": "Mathematics.",
        "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.",
        "rationale": "Context.",
        "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"
                  }
                ]
              }
            ]
          }
        ]
      }
    ]
  },
  "openQuestions": [
    "Should abundancy index be a separate entry?",
    "How should number theory references be linked?",
    "How should the status of unresolved numbers be tracked?"
  ],
  "statistics": {
    "bundles": 4,
    "layers": 8,
    "findings": 8,
    "questions": 16
  }
}