{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R1315",
  "content_hash": null,
  "slug": "grahl-sequence-infinitely-many-ones-status-20260801",
  "type": "claim",
  "title": "Current checked status and unresolved remainder",
  "summary": "UNKNOWN as of 2026-07-27. The MathOverflow thread remains open with zero answers, and the checked OEIS records still present the infinitude of ones as an unresolved question. No later proof or counterexample was located.",
  "relevance": "Records the strongest checked neighboring results and the exact remainder future work must settle.",
  "relevance_source": "recorded",
  "body": "A dated independent review on 2026-08-01 checked the structured sources below, the complete visible source discussion, exact-title and equivalent-formulation searches, and the current TheoremDB corpus. On 2026-07-27 the Stack Exchange API reported zero answers, no accepted answer, and no closure for MathOverflow question 338415; every visible comment was checked, including the proof for a different weakly-increasing variant. The dated search checked OEIS A229037, which defines the same greedy sequence, and OEIS A236246, which records the observed indices of the value 1. Neither entry supplied a proof of infinitude. A TheoremDB corpus search for A229037, Grahl sequence, greedy arithmetic-progression sequence, and the exact recurrence found no existing target. Reusable work consists of certified prefix blocks, occurrence indices, and recurrence-state summaries. Evidence for the weakly-increasing variant does not settle this sequence.\n\nA complete resolution must satisfy: Prove that for every integer N there exists m>N with a_m=1, or prove that a_m is never 1 beyond an explicit index. Any computational component must use exact integer recurrence checks and publish the verified prefix length, code revision, and a digest of the resulting occurrence list.",
  "status": "reported",
  "evidence_grade": "sourced",
  "scope": null,
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://mathoverflow.net/questions/338415/on-the-first-sequence-without-triple-in-arithmetic-progression",
      "locator": "Dataset references and independent 2026-08-01 status search."
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://mathoverflow.net/questions/338415/on-the-first-sequence-without-triple-in-arithmetic-progression",
    "locator": "Dataset references and independent 2026-08-01 status search."
  },
  "models": [],
  "relations": [
    {
      "slug": "R1314",
      "title": "Complete the stated acceptance conditions",
      "object_type": "attempt",
      "relation": "addresses",
      "direction": "incoming"
    },
    {
      "slug": "R1573",
      "title": "Dated status and exact unresolved remainder",
      "object_type": "claim",
      "relation": "supersedes",
      "direction": "incoming",
      "metadata": {
        "reason": "Replaces unreadable status prose with the dated review from 2026-08-01."
      }
    },
    {
      "slug": "grahl-sequence-infinitely-many-ones",
      "title": "grahl sequence infinitely many ones",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}