Problem 423

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claims/: The 2 claim pages of Problem 423, one per claimant's result; the problem's standing derives from them.

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Statement. Let $a_1=1$ and $a_2=2$ and for $k\geq 3$ choose $a_k$ to be the least integer $>a_{k-1}$ which is the sum of at least two consecutive terms of the sequence. What is the asymptotic behaviour of this sequence?

Status. Open. The site labels the problem OPEN (page last edited 23 March 2026) and credits Bolan and Tang [Ta26], independently, with the sequence omitting infinitely many integers, and Tang with the bounds $n+\Omega(\log\log n)\le a_n\ll n^{688/413+o(1)}$; the omission result and Tang's bounds with upper exponent $4175/2506$ (the exponent $688/413$ is Sothanaphan's observation recorded in the paper, not one of its theorems) are pending partial claims on Tang's page and Bolan's page. The standing in the frontmatter derives from the claim pages.

Source. erdosproblems.com/423, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #423, https://www.erdosproblems.com/423.

References.

• [Cu25] A. Cushman, A Note on the Sum-Product Problem and the Convex Sumset Problem. arXiv:2512.13849 (2025).
• [Er77c] Erdős, Paul, Problems and results on combinatorial number theory. III. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976) (1977), 43-72.
• [Ta26] Q. Tang, The Hofstadter consecutive-sum sequence omits infinitely many positive integers. arXiv:2603.09939 (2026).

Formalization. Statement in formal-conjectures, linked at its commit of 18 September 2026, with sorry; the file marks the main statement research open and five variants research solved, as the Current assessment records.

Current assessment

The question, as the site states it (page last edited 23 March 2026): what is the asymptotic behavior of the sequence $1,2,3,5,6,8,10,11,\ldots$ (OEIS A005243) in which each term is the least integer above the last that is a sum of at least two consecutive earlier terms? It is open. The known bounds are

$$n+\Omega(\log\log n)\le a_n\ll n^{688/413+o(1)},$$

from Tang's arXiv paper [Ta26]: its Theorem 1.4 is the lower bound, its Theorem 1.5 gives $a_n\ll n^{4175/2506+o(1)}$ from Bloom's lower bound for difference sets of convex sets, and its Section 6.2 records Sothanaphan's observation that Cushman's sharper convex-set bound [Cu25] improves the exponent to $688/413<1.6659$ by the same argument. The qualitative statement that $a_n-n$ is nondecreasing and unbounded, so that the sequence omits infinitely many positive integers, was proved independently by Tang (note of 15 January 2026, Theorem 1.3 of the paper) and by Bolan (note of 15 January 2026, with the explicit form that every interval $[N,2^{3N+2}]$ misses some integer); the two claim pages record the results. The conjecture of Erdős and Hegyvári that convex sets satisfy $|A-A|\ge|A|^{2-o(1)}$ would give $a_n\le n^{1+o(1)}$, and the site and Tang's paper expect $a_n=n+o(n)$; neither is proved. None of the results is refereed, and the site's label is OPEN, so both claims are pending. The formal-conjectures statement file (linked at its commit in the Formalization line) marks the main statement $a_n=n+o(n)$ research open and five variants research solved: the nondecreasing, unbounded and infinite-complement variants credit Bolan and Tang, and the upper-bound and lower-bound variants credit Tang; all have sorry bodies, and the community database lists the problem as open with its statement formalized and no formal proof. The thread's other posts (numerical plots of $a_n-n$ up to $n=30000$, suggesting growth like $n^\alpha$ with $1/5<\alpha<1/2$, and discussion of the AI systems used) are comments without a manuscript and have no pages.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• tang_2026_hofstadter_consecutive_sum_sequence_omits_infinitely
• tang_2026_hofstadter_consecutive_sum_sequence_omits_infinitely / lemma_2_1
• tang_2026_hofstadter_consecutive_sum_sequence_omits_infinitely / theorem_1_3
• tang_2026_hofstadter_consecutive_sum_sequence_omits_infinitely / theorem_1_4
• tang_2026_hofstadter_consecutive_sum_sequence_omits_infinitely / theorem_1_5