{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R650",
  "content_hash": null,
  "slug": "rdcr-claim-literature-audit",
  "type": "claim",
  "title": "Nearby literature treats different consecutive-divisor questions",
  "summary": "The audited sources cover equal tau values, prescribed tau progressions, distinct omega values, and ratios of adjacent tau values; none supplies this bounded rainbow record.",
  "relevance": "For Longest rainbow divisor-count interval below 10^12, record rdcr-claim-literature-audit (“Nearby literature treats different consecutive-divisor questions”) records a bound, answer, status fact, or structural consequence. The record states: The audited sources cover equal tau values, prescribed tau progressions, distinct omega values, and ratios of adjacent tau values; none supplies this bounded rainbow record.",
  "relevance_source": "recorded",
  "body": "Letsko studies consecutive integers having one fixed value of \\(\\tau\\), including long runs with 12 or 24 divisors. De Koninck, Friedlander, and Luca prove existence bounds for consecutive integers whose \\(\\omega\\) or \\(\\Omega\\) values are pairwise distinct. Those functions count prime factors rather than divisors. Eberhard proves that every positive rational occurs infinitely often as \\(\\tau(n+1)/\\tau(n)\\), a result about adjacent pairs.\n\nOEIS A363335 catalogs runs satisfying the prescribed pattern \\(\\tau(m+j)=2(n+j)\\). Its rows give examples with distinct divisor counts under an extra linear constraint. A targeted search of these papers, their references, arXiv, and OEIS did not locate a published maximum for arbitrary pairwise-distinct \\(\\tau\\)-values below \\(10^{12}\\). This is a scoped status report rather than a proof of novelty.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "a targeted literature and sequence-database search for consecutive integers with pairwise distinct values of the ordinary divisor function, checked through 2026-07-25",
    "bounds": {
      "status_year": {
        "min": 2026,
        "max": 2026
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1510.07081",
      "locator": "Vladimir A. Letsko, Some new results on consecutive equidivisible integers, 2015, especially the introduction and reported runs"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1510.07081",
    "locator": "Vladimir A. Letsko, Some new results on consecutive equidivisible integers, 2015, especially the introduction and reported runs"
  },
  "relations": [
    {
      "slug": "R649",
      "title": "The requested maximum is at least fourteen",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "rainbow-divisor-count-run-1e12",
      "title": "rainbow divisor count run 1e12",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}