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Claim. The number

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

is irrational: the case $k=1$ of Problem 252, as a specialization of two general theorems of P. Erdős and E. G. Straus. Theorem 3.7 of On the irrationality of certain series, Pacific J. Math. 55 (1974), no. 1, 85--92 (printed pp. 88--89; the second paper link), states that if the positive integers $a_n$ are monotonic and, for some $\delta>0$, exceed $n^{1/2+\delta}$ for large $n$, then $1$, $\sum\varphi(n)/(a_1\cdots a_n)$ and $\sum\sigma(n)/(a_1\cdots a_n)$, together with a series with small integer numerators, are linearly independent over $\mathbb Q$; the sequence $a_n=n$, with $a_1=1$, meets these hypotheses, so $\sum\sigma(n)/n!$ is irrational, as the card Theorem 3.7 records. Theorem 2.26 of Some number theoretic results, Pacific J. Math. 36 (1971), no. 3, 635--646 (printed p. 642; the first link), gives the irrationality earlier: for a monotonic integer sequence with $a_n\ge n^{11/12}$ for large $n$, the series $\sum\sigma(n)/(a_1\cdots a_n)$ and $\sum\varphi(n)/(a_1\cdots a_n)$ are irrational. That section's standing convention $2\le a_1$ is not met by $a_1=1$, but the proof uses the $a_n$ only at large $n$, so reading the theorem at $a_n=n$ is this compilation's reading, recorded on the card erdos_1971_number_theoretic_results. Neither paper states the $n!$ case itself. Schlage-Puchta 2006 attributes the cases $k=0$ and $k=1$ to the 1974 paper, and formal-conjectures tags its variant erdos_252.variants.k_eq_one research solved, crediting the same paper. The same 1971 section's Lemma 2.14 with $a_n=n$ gives the divisor-count series $\sum d(n)/n!$ irrational, the variant $k=0$ outside the question. The Monthly solution of the case $k=1$ is Erdős's claim page.

Covers. The case $k=1$ only: $\sum_{n\ge1}\sigma_1(n)/n!$ is irrational. Nothing about any $k\ge2$.

Acceptance. Refereed: the Pacific Journal of Mathematics, volume 36, issue 3 (March 1971), pp. 635--646, and volume 55, issue 1 (November 1974), pp. 85--92; the Crossref records of the DOIs give these data. The site labels the problem OPEN, so no curator credit is acceptance and no reviewed evidence is listed. The proofs are not checked here.

Depends on. Nothing in this wiki.