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Claim. The number

$$\alpha_2=\sum_{n\ge1}\frac{\sigma_2(n)}{n!},$$

with $\sigma_2(n)$ the sum of the squares of the divisors of $n$, is irrational: the case $k=2$ of Problem 252. Erdős and Kac posed it as Problem 4518 in the problem section of the American Mathematical Monthly, volume 60 (1953), no. 1, p. 47 (the first paper link is the Monthly's problems item on that page), and the published solution is R. Breusch's, volume 61 (1954), no. 4, pp. 264--265 (the second link). Erdős's 1957 paper (remark on p. 213, footnote 2) attributes the case $k=2$ to Breusch's solution; Erdős's 1988 survey (card, p. 102) writes that he and Kac proved the cases $k=1$ and $k=2$ and cites this Monthly problem for both; Schlage-Puchta 2006 attributes the case $k=2$ to Erdős and Kac, Friedlander, Luca and Stoiciu 2007 attribute both cases to this problem, and Pratt 2022 identifies its solution as the source of the case $k=2$. formal-conjectures tags its variant erdos_252.variants.k_eq_two research solved, citing the same item. The Monthly items are paywalled and not held in the library, so the published argument is recorded by these attributions; the site's remarks credit Erdős with both cases under the key of Problem 4493.

Covers. The case $k=2$ only: $\sum_{n\ge1}\sigma_2(n)/n!$ is irrational. Nothing about any $k\ge3$.

Acceptance. Refereed: the American Mathematical Monthly, volume 61, issue 4 (April 1954), pp. 264--265, carries Breusch's solution of Problem 4518; the Crossref record of its 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.