Problem 18

We call $m$ practical if every integer $1\leq n<m$ is the sum of distinct divisors of $m$ . If $m$ is practical then let $h(m)$ be such that $h(m)$ many divisors always suffice.

Are there infinitely many practical $m$ such that

h(m) < (\log\log m)^{O(1)}?

Is it true that $h(n!)<n^{o(1)}$ ? Or perhaps even $h(n!)<(\log n)^{O(1)}$ ?

Source: erdosproblems.com/18

Number theory
Divisors
Factorials

Status

Open

No claim settles this problem.

OPEN on the site (page last edited 11 April 2026), with the prize attached to the first question. The frontmatter standing open derives from the claim pages: the accepted partial claim JenW1N's Lean proof settles the second question, $h(n!)<n^{o(1)}$ ; the first question carries two pending partial claims, Price's and van Doorn and GPT-6 Astra Pro's; the third question, $h(n!)<(\log n)^{O(1)}$ , has no claim. The claim pages and the assessment below carry the evidence for each part. The site's wording attaching the prize to the first question was settled on 11 April 2026 after wavering; the assessment gives the history.

Er81h: Erdős, P., Some problems and results on additive and multiplicative number theory. Analytic number theory (Philadelphia, Pa., 1980) (1981), 171-182.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève, Geneva, 1980; pp. 37--38 (cited from Hughes 2026 for $h(n!)\le n$ and the two factorial questions).
Vo85: Vose, Michael D., Egyptian fractions. Bull. London Math. Soc. (1985), 21-24.
Vo84: Vose, M. D., Integers with consecutive divisors in small ratio. J. Number Theory 19 (1984), no. 2, 233--238 (the van Doorn note's citation for the $(\log m)^{1/2}$ construction, where the site cites [Vo85]; the attribution is not resolved).
Yo92: Yokota, H., On a sum of divisors. Canad. Math. Bull. 35 (1992), no. 3, 423--430 (Corollary 1, cited from the van Doorn note).
TeYo90: Tenenbaum, G. and Yokota, H., Length and denominators of Egyptian fractions, III. J. Number Theory 35 (1990), 150--156 (Lemma 4; cited from Hughes 2026).
Yo95: Yokota, H., On a knapsack problem involving divisors of $n!$ . Res. Bull. Hiroshima Inst. Tech. 29 (1995), 25--28 (cited from Hughes 2026).
Hu26: Hughes, S. D., Sums of distinct divisors of factorials. arXiv:2609.10902 (v1, 9 September 2026); card Hughes 2026.

Problem 18

Status

Proof claims

Discussion