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Claim. Part (2) 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

$$S_3=\sum_{n\ge1}\frac{\sigma_3(n)}{n!}$$

is irrational, with no hypothesis: the case $k=3$ of Problem 252. The argument assumes $S_3=a/b$, so that $(n-1)!\,S_3$ forces the tail $\sum_{\nu\ge n}\sigma_3(\nu)/(\nu)_{\nu-n+1}$ to be an integer for large $n$, and takes $n$ among the primes $q\le x$ for which the least prime factors of $(q+1)/2$ and $(q+2)/3$ both exceed $x^{1/9}$, of which a sieve lemma that the paper derives from Halberstam and Richert (Theorem 7.4) supplies at least order $x/\log^3x$. For such $q$ the fractional parts of $\sigma_3(q)/q$, of $\sigma_3(q+2)/(q(q+1)(q+2))$ and of $\sigma_3(q+1)/(q(q+1))-\sigma_3(q+1)/(q+1)^2$ are fixed up to $O(q^{-1/3})$, so integrality confines $\sigma_3(q+1)/(q+1)^2=9\sigma_3(n)/(4n^2)$, with $n=(q+1)/2$, to within $O(n^{-1/3})$ of a fixed residue modulo 1 for at least order $x/\log^3x$ integers $n\le x$; the paper bounds the number of such $n$ by $O(x\log\log x/\log^4x)$ through the Erdős–Turán inequality, van der Corput estimates and a sieve count, a contradiction (the arXiv text prints the constants of this step as $7/8$ and $19/216$, where direct computation gives $1/8$ and $35/216$; the argument does not depend on them). The source card schlagepuchta_2006_irrationality_number_theoretical_series names the arXiv posting of 2011 (the preprint link) as the copy read, whose first page states the theorem; no file is held. Part (1) of the same Theorem, the irrationality of every $S_k$ under Schinzel's Hypothesis H, is the conditional claim on its own page. Friedlander, Luca and Stoiciu proved the same case independently in 2007, as their note added in March 2007 records (their claim page), and formal-conjectures tags its variant erdos_252.variants.k_eq_three research solved, citing both papers.

Covers. The case $k=3$ only: $\sum_{n\ge1}\sigma_3(n)/n!$ is irrational. Nothing unconditional about any $k\ge4$.

Acceptance. Refereed: the Ramanujan Journal, volume 12, issue 3 (December 2006), pp. 455--460; the Crossref record of the DOI gives these data, and the journal version was not compared with the arXiv posting. The site labels the problem OPEN, so its curator's remark crediting this paper and Friedlander, Luca and Stoiciu with the case $k=3$ 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.