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Claim. The number

$$\alpha_1=\sum_{n\ge1}\frac{\sigma(n)}{n!},$$

with $\sigma(n)=\sigma_1(n)$ the sum of the divisors of $n$, is irrational: the case $k=1$ of Problem 252. Erdős posed it as Problem 4493 in the problem section of the American Mathematical Monthly, volume 59 (1952), p. 412 (the first paper link is the Monthly's problems item on that page), and the published solution is J. B. Kelly's, volume 60 (1953), no. 8, p. 557 (the second link). Erdős's 1957 paper (remark on p. 213, footnote 2) attributes the case $k=1$ to Kelly's solution and the case $k=2$ to Breusch's, and Pratt's 2022 paper (its references [9] and [11]) identifies the solutions of Problems 4493 and 4518 as the sources of the two cases; the site's remarks credit Erdős with both cases under its key [Er52], Erdős's 1988 survey writes that he and Kac proved them, and Friedlander, Luca and Stoiciu attribute both to Problem 4518. The Monthly items are paywalled and not held in the library, so the published argument is recorded by these attributions. formal-conjectures tags its variant erdos_252.variants.k_eq_one research solved, citing the 1974 paper of Erdős and Straus, whose general theorems give the same case on their own claim page.

Covers. The case $k=1$ only: $\sum_{n\ge1}\sigma_1(n)/n!$ is irrational. Nothing about any $k\ge2$; the case $k=2$ is Erdős and Kac's.

Acceptance. Refereed: the American Mathematical Monthly, volume 60, issue 8 (October 1953), p. 557, carries Kelly's solution of Problem 4493; the Crossref record of the DOI gives these data. The site labels the problem OPEN, so its curator's remark crediting Erdős with the cases $k=1$ and $k=2$ 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.