Erdős Problem #423 — open problem in combinatorics

Erdős Problem #423

Erdős Problem 423 [Er77c, p.71; ErGr80, p.83]: Let a(1) = 1, a(2) = 2, and for k ≥ 3 let a(k) be the least integer greater than a(k-1) that is a sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence? It seems likely that a_n = n + o(n).

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-423,
  title        = {Erdős Problem #423},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-423}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Erdős Problem 423 [Er77c, p.71; ErGr80, p.83]:

Let $a (1) = 1$ , $a (2) = 2$ , and for $k \geq 3$ let $a (k)$ be the least integer greater than $a (k - 1)$ that is a sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence? It seems likely that $a_{n} = n + o (n)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«423». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_423 : answer(sorry) ↔
    ∀ a : ℕ → ℕ, IsHofstadterSeq a →
    (fun n : ℕ => (a n : ℝ) - n) =o[atTop] (fun n : ℕ => (n : ℝ))

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/423. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/423
[Er77c] Erdős, P., Problems and results on combinatorial number theory. III, Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), 1977, pp. 43–72.
[ErGr80] Erdős, P. and Graham, R., *Old and new problems and results in combinatorial number theory*, Monographies de L'Enseignement Mathématique (1980).
[Cu25] Cushman, A., A Note on the Sum-Product Problem and the Convex Sumset Problem. arXiv:2512.13849 (2025).
[Ta26] Tang, Q., *The Hofstadter consecutive-sum sequence omits infinitely many positive integers*. arXiv:2603.09939 (2026).
[Bolan] Bolan, M., Hofstader–Ulam Sequence, https://github.com/mjtb49/HofstaderUlam/blob/main/HofstaderUlamSequence.pdf
OEIS A005243

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.