{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R166",
  "content_hash": null,
  "slug": "csgf-claim-oeis-a055396-transform",
  "type": "claim",
  "title": "The Grundy sequence is a shifted A055396",
  "summary": "For n greater than 1, the game value is one below the index of the least prime divisor.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "OEIS A055396 defines \\(a(1)=0\\) and, for \\(n>1\\), \\(a(n)=\\pi(P(n))\\), where \\(P(n)\\) is the least prime divisor. The theorem in this entry gives\n\\[\n g(1)=0,\\qquad g(n)=a(n)-1\\quad(n>1).\n\\]\nThe initial Grundy values are\n\\[\n0,0,1,0,2,0,3,0,1,0,4,0,5,0,1,0,6,0,7,0,1,0,8,0,2,0,1,0,9,0.\n\\]\nIannucci and Larsson prove in their Theorem 5 that A055396 itself is the Grundy sequence of the totative game, which moves a single integer to a smaller coprime residue. The splitting game here replaces a heap by a disjunctive sum, so its recurrence is different even though the same least-prime obstruction controls the mex.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the full Grundy sequence of the unequal coprime splitting game"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://oeis.org/A055396",
      "locator": "OEIS A055396, definition, comments, and link to Iannucci-Larsson; accessed 2026-07-24"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://oeis.org/A055396",
    "locator": "OEIS A055396, definition, comments, and link to Iannucci-Larsson; accessed 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R167",
      "title": "The least-prime-factor formula holds for every heap",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "R165",
      "title": "Exact recurrence sweep through 30,000",
      "object_type": "artifact",
      "relation": "tests",
      "direction": "incoming"
    },
    {
      "slug": "coprime-splitting-grundy-formula",
      "title": "coprime splitting grundy formula",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}