Problem 252

Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$ . Is

\sum \frac{\sigma_k(n)}{n!}

irrational?

Source: erdosproblems.com/252

Number theory
Irrationality

Wiki page

Status

SolvedProved

An accepted solution exists. The statement is true.

Open on the site: the site labels the problem OPEN (page last edited 2026-01-22), the community database and formal-conjectures list it open, and nobody outside this repository had examined or acknowledged a proof by that date. The frontmatter standing solved, proved derives from the claim page (Tokengrinder, 2026) of a kernel-checked Lean 4 proof published anonymously in September 2026, which answers yes for every $k\ge1$ (and covers $k=0$ ); its only acceptance evidence is formalized: the proof was rebuilt and kernel-replayed here and its whole statement was found faithful by a graded fresh-context review, the third-round fidelity review of 2026-09-18 and its distinct grade, which docs/anatomy.md counts as documented independent acceptance of the external result; the Current assessment states what the two earlier review rounds found and why their grades warrant nothing. There is no refereed write-up, and the mechanical facts trust Lean's kernel and the consistency of Mathlib v4.33.1. Refereed work settles $1\le k\le4$ unconditionally (Erdős, Erdős–Straus and Erdős–Kac for $k=1,2$ ; Schlage-Puchta 2006 and Friedlander–Luca–Stoiciu 2007 for $k=3$ ; Pratt, Acta Arith. 211 (2023) for $k=4$ ) and every $k$ under Schinzel's Hypothesis H or Dickson's conjecture; each of these results has an accepted partial or conditional claim page, listed under Known Results.

ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monogr. Enseign. Math. 28 (1980), p. 62, item (ii).
Er88c: Erdős, P., On the irrationality of certain series: problems and results. New advances in transcendence theory (Durham, 1986), Cambridge Univ. Press (1988), 102–109; p. 102.
Er52: Erdős, P., Problem 4493. Amer. Math. Monthly 59 (1952), 412; solution J. B. Kelly, 60 (1953), 557–558. Paywalled and not held; the site's entry gives the solution's pages under the 1952 date. Claim page: the case k = 1.
ErKa54: Erdős, P. and Kac, M., Problem 4518. Amer. Math. Monthly 60 (1953), 47; solution R. Breusch, 61 (1954), 264–265. Paywalled and not held. Claim page: the case k = 2.
Er57: Erdős, P., On the irrationality of certain series. Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), 212–219; p. 213.
ErSt71: Erdős, P. and Straus, E. G., Some number theoretic results. Pacific J. Math. 36 (1971), 635–646.
ErSt74: Erdős, P. and Straus, E. G., On the irrationality of certain series. Pacific J. Math. 55 (1974), 85–92.
ScPu06: Schlage-Puchta, J.-C., The irrationality of a number theoretical series. Ramanujan J. 12 (2006), no. 3, 455–460; arXiv:1105.1452.
FLC07: Friedlander, J. B., Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers 7 (2007), #A31, 9 pp.
Pr22: Pratt, K., The irrationality of a divisor function series of Erdős and Kac. Acta Arith. 211 (2023), no. 3, 193–228; arXiv:2209.11124 (2022).
Gu04: Guy, R. K., Unsolved problems in number theory. 3rd ed., Problem Books in Mathematics, Springer (2004), xviii+437 pp. Section B14 "Some irrational series", printed p. 104: "Is $\sum_{n=1}^\infty(\sigma_k(n)/n!)$ irrational? It is for $k=1$ and 2", then Erdős's $\sum1/(2^n-1)=\sum d(n)/2^n$ and Borwein's series; its reference list (p. 105) includes Erdős and Kac's Monthly Problem 4518 with Breusch's solution, but not Problem 4493. Library home: Guy 2004.
DeSi11: Deajim, A. and Siksek, S., On the $\mathbb{Q}$ -linear independence of the sums $\sum_{n\ge1}\sigma_k(n)/n!$ . J. Number Theory 131 (2011), no. 4, 745–749. Paywalled and not held; a related conditional result. - Tokengrinder, Erdős 252: irrationality of the factorial divisor-sum series. Lean 4 development, https://github.com/tokengr1nder/Erdos252 (2026); the reviewed commit of 2026-09-13 is pinned on the claim page.

Problem 252

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