# P2872: Infinitely many ones in the greedy three-term-progression-free sequence
- ID: `P2872`
- Reference: `grahl-sequence-infinitely-many-ones`
- Page: https://theoremdb.org/statements/P2872
- Record maturity: Reviewed problem with recorded work
## Problem
Define a sequence \((a_m)_{m\ge 1}\) of positive integers recursively by \(a_1=1\), and, for each \(m\ge 2\), let \(a_m\) be the least positive integer such that \(a_{m-2k}+a_m\ne 2a_{m-k}\) for every integer \(k\) with \(1\le k\le (m-1)/2\). Is the set \(\{m\ge 1:a_m=1\}\) infinite?
### Problem setup
- **Remark.** The recurrence is greedy in lexicographic order: at position m it chooses the smallest positive value that creates no three-term arithmetic progression in the values at equally spaced indices ending at m.
- **Definition.** A three-term arithmetic progression here means indices m-2k, m-k, m together with values satisfying a_{m-2k}+a_m=2a_{m-k}; the three values need not be distinct.
### What counts as a solution
- 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
Unresolved in this packet after the dated source check. Strongest checked result: 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. Exact unresolved remainder: 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. [1](#reference-1) [2](#reference-2) [3](#reference-3)
## Work
### Evidence for the current status
**Claim 1 (Dated status and exact unresolved remainder).** Unresolved in this packet after the dated source check. Strongest checked result: 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. Exact unresolved remainder: 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.
The packet's cited sources and equivalent formulations were checked in the dated review recorded below.
Strongest checked result: 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.
Exact unresolved remainder: 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.
### Background and intake notes
The sequence is forced by a local greedy rule but displays large sparse regions. Long prefix computations are easy to duplicate, so certified occurrence lists and failed structural recurrences are useful shared artifacts.
- Original intake status: 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.
- 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.
- Recorded example: The sequence begins 1,1,2,1,1,2,2,4,4,1, and hence the first positions carrying 1 are 1,2,4,5,10.
### Open directions
- **Route 1** (reported): 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. [1](#reference-1)
### Computational notes
- OEIS A236246 records a long prefix of the indices m for which a_m=1, but a finite list cannot decide infinitude.
### Working on this
Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `grahl-sequence-infinitely-many-ones`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.
## References
1. MathOverflow question 338415, “Infinitely many ones in the greedy three-term-progression-free sequence,” checked 2026-08-01. Question 338415 and all visible comments, checked through the Stack Exchange API on 2026-07-27. https://mathoverflow.net/questions/338415/on-the-first-sequence-without-triple-in-arithmetic-progression
- Also cited at Full question, answers, and visible comments concerning Infinitely many ones in the greedy three-term-progression-free sequence; checked 2026-08-01.
- Also cited at Editorial research route recorded 2026-08-01.
- forum; reference source; checked 2026-08-01
- Source use: citation_only
- Source used to formulate or check the problem record.
- Source used to assess the problem's recorded status.
- For Infinitely many ones in the greedy three-term-progression-free sequence: This is an original CC0 textbook restatement motivated by the cited MathOverflow thread; no MathOverflow prose was copied.
- Source named by the research packet.
2. OEIS Foundation Inc., entry A229037, checked 2026-08-01. definition, terms, formulas, programs, and linked million-term tables for the forest-fire sequence https://oeis.org/A229037
- reference_database; reference source; checked 2026-08-01
- Source use: citation_only
- Reused material: definition, terms, formulas, programs, and linked million-term tables for the forest-fire sequence
- Reuse basis: fair_use_reviewed; rights holder: The OEIS Foundation Inc. and the credited contributors; checked 2026-08-01; by Philip Weiss, TheoremDB staff
- Required attribution: OEIS Foundation Inc., entry A229037, checked 2026-08-01.
- Source used to assess the problem's recorded status.
- For Infinitely many ones in the greedy three-term-progression-free sequence, this source defines the exact greedy sequence and supplies reproducible prefix data.
3. OEIS Foundation Inc., entry A236246, checked 2026-08-01. definition as indices n with A229037(n)=1, listed terms, infinitude comment, and exact programs https://oeis.org/A236246
- reference_database; reference source; checked 2026-08-01
- Source use: citation_only
- Reused material: definition as indices n with A229037(n)=1, listed terms, infinitude comment, and exact programs
- Reuse basis: fair_use_reviewed; rights holder: The OEIS Foundation Inc. and the credited contributors; checked 2026-08-01; by Philip Weiss, TheoremDB staff
- Required attribution: OEIS Foundation Inc., entry A236246, checked 2026-08-01.
- Source used to assess the problem's recorded status.
- For Infinitely many ones in the greedy three-term-progression-free sequence, this source records the exact support sequence and explicitly preserves the infinitude question.