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Claim. Part (1) of the single Theorem of Jan-Christoph Schlage-Puchta, The irrationality of a number theoretical series, Ramanujan J. 12 (2006), no. 3, 455--460, states that if Schinzel's Hypothesis H holds then

$$S_k=\sum_{n\ge1}\frac{\sigma_k(n)}{n!}$$

is irrational for every $k\in\mathbb N$, which would answer Problem 252 yes for every $k\ge1$. The claim is conditional: Hypothesis H asserts that finitely many irreducible integer polynomials with positive leading coefficients, whose product has no fixed prime divisor, take prime values simultaneously at infinitely many integers, and it is unproven, so this page derives nothing for the problem's standing. The argument assumes $S_k=a/b$, so that $(n-1)!\,S_k$ forces the tail $\sum_{\nu\ge n}\sigma_k(\nu)/(\nu)_{\nu-n+1}$ to be an integer, and takes $n$ among the primes $q\equiv1\pmod{k!^k}$ for which $(q+i)/(i+1)$ is prime for all $i\le k$, which Hypothesis H supplies; two incompatible estimates of the distance from the tail to the nearest integer end in the contradiction $\|q_1^{k-1}/p^k\|<q_1^{-1+\varepsilon}$ for arbitrarily large $q_1$ with $p$ fixed. The source card schlagepuchta_2006_irrationality_number_theoretical_series names the arXiv posting of 2011 (the preprint link) as the copy read; no file is held. The unconditional part (2) of the Theorem, the case $k=3$, is the partial claim on its own page; Friedlander, Luca and Stoiciu proved a parallel conditional theorem under Dickson's conjecture (their conditional page). formal-conjectures tags its variant erdos_252.variants.schinzel, which takes Hypothesis H as an explicit hypothesis, research solved, citing this paper.

Acceptance. Refereed: the Ramanujan Journal, volume 12, issue 3 (December 2006), pp. 455--460; the Crossref record of the DOI gives these data. The site labels the problem OPEN, so its curator's remark that the sum is irrational for every $k$ under Schinzel's conjecture by this paper is commentary on an open problem and not acceptance, and no reviewed evidence is listed. The proof is not checked here.

Depends on. Nothing in this wiki.