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Claim. Theorem 1 of Kyle Pratt, The irrationality of a divisor function series of Erdős and Kac, Acta Arith. 211 (2023), no. 3, 193--228 (arXiv:2209.11124, posted 2022-09-22), states that

$$\alpha_4=\sum_{n\ge1}\frac{\sigma_4(n)}{n!}=42.30104\ldots$$

is irrational, with no hypothesis: the case $k=4$ of Problem 252. The proof is sieve-theoretic, combined with exponential sum estimates, and the author writes that it pushes those techniques to their limit, so that new ideas seem necessary for $k\ge5$. The paper also frames $\alpha_k$ through the entire $E$-functions $f_k(z)=\sum\sigma_k(n)z^n/n!$ and remarks that they do not appear to satisfy any differential equation to which the Siegel–Shidlovskii technique applies. The source card pratt_2022_irrationality_divisor_function_series_erdos_kac names the arXiv posting (the preprint link) as the copy read, whose first page states the theorem, and holds no file; the journal version is paywalled and was not compared. formal-conjectures tags its variant erdos_252.variants.k_eq_four research solved, citing the arXiv posting.

Covers. The case $k=4$ only: $\sum_{n\ge1}\sigma_4(n)/n!$ is irrational. Nothing about any $k\ge5$, which the paper leaves open.

Acceptance. Refereed: Acta Arithmetica, volume 211, issue 3 (2023), pp. 193--228; the Crossref record of the DOI gives these data. The site labels the problem OPEN, so its curator's remark crediting this paper with the case $k=4$ 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.