Problem 1054

Let $f(n)$ be the minimal integer $m$ such that $n$ is the sum of the $k$ smallest divisors of $m$ for some $k\geq 1$ .

Is it true that $f(n)=o(n)$ ? Or is this true only for almost all $n$ , and $\limsup f(n)/n=\infty$ ?

Number theory
Divisors

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ClaimedAnswered

A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Open. The site labels the problem OPEN (page last edited 6 December 2025). Its commentary credits Tao, in comments first posted under Problem 468, with disproving the strong claim $f(n)=o(n)$ by showing that the $n$ with $f(n)\le\delta n$ have upper density $\ll\delta^2$ . The frontmatter standing derives from the claim pages: two pending full claims answer all three parts, so it is claimed.

Gu04: Guy, Richard K., Unsolved problems in number theory. Third edition, Problem Books in Mathematics, Springer, New York (2004), xviii+437 pp.; doi:10.1007/978-0-387-26677-0. Section B2 "Almost perfect, quasi-perfect, pseudoperfect, harmonic, weird, multiperfect and hyperperfect numbers", printed p. 80, where the book states the question and tabulates $f(n)$ for $n\le28$ . Library home: Guy 2004.

Problem 1054

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