Problem 359

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Statement. Let $a_1<a_2<\cdots$ be an infinite sequence of integers such that $a_1=n$ and $a_{i+1}$ is the least integer which is not a sum of consecutive earlier $a_j$s. What can be said about the density of this sequence?

In particular, in the case $n=1$, can one prove that $a_k/k\to \infty$ and $a_k/k^{1+c}\to 0$ for any $c>0$?

Formulation. Read as the site words it, the rule fails for $n\ge2$. When $a_2$ is chosen, the only sum of consecutive earlier terms is $a_1=n$, so the least positive integer that is not such a sum is $1$, giving $a_2<a_1$ against $a_1<a_2<\cdots$; no sequence meets the site's wording. Erdős and Graham (1980, p. 59) use the same wording, with $a_1=k$. Formal-conjectures reads $a_{i+1}$ as the least integer exceeding $a_i$ that is not such a sum, and the questions are recorded under that reading. For $n=1$ the two readings agree, since every positive integer up to $a_i$ is already such a sum when $a_{i+1}$ is chosen.

Status. Open.

Source. erdosproblems.com/359, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #359, https://www.erdosproblems.com/359.

References.

• [An75] Andrews, George E., Research Problems: Mac Mahon's Prime Numbers of Measurement. Amer. Math. Monthly (1975), 922-923.
• [Po77] Porubský, Š., On MacMahon's segmented numbers and related sequences. Nieuw Arch. Wisk. (3) 25 (1977), 403-408.

Formalization. Statement in formal-conjectures.

Current assessment

The standing recorded here targets the site's wording (page last edited 28 December 2025), whose rule for $a_{i+1}$ fails as written for $n\ge2$ and is read as the Formulation above states. No independent assessment of proof coverage is recorded, and the problem has no claim page. The site's commentary credits Porubský [Po77] with two results for $n=1$: for every $\varepsilon>0$ infinitely many $k$ have $a_k<(\log k)^\varepsilon k\log k/\log\log k$, and $\limsup A(x)/\pi(x)\ge1/\log2$, where $A(x)$ counts the terms up to $x$. Neither decides the limits the problem asks about: the first gives only $\liminf_k a_k/k^{1+c}=0$, and the second bounds $A(x)$ from below only along a sequence of $x$. They settle no instance, so they have no claim page. Andrews [An75] conjectures $a_k\sim k\log k/\log\log k$.

Search scope. As of 2026-10-07 the site's page lists no proof claim, its discussion thread holds one comment, of 29 November 2025, restating Porubský's first result in its $\liminf$ form, and the formal-conjectures statement file states both limits for $n=1$ as open and has no solved variant.

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.

• erdos_1977_problems_results_combinatorial_number_theory_iii
• erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk
• erdos_1980_noen_mindre_kjente_problemer_i_kombinatorisk / question_p160