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Claim. Theorem 1 of John B. Friedlander, Florian Luca and Mihai Stoiciu, On the irrationality of a divisor function series, Integers 7 (2007), #A31, 9 pp., states that

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

is irrational, with no hypothesis: the case $k=3$ of Problem 252. The argument is the classical one: assume the sum is $A/B$, multiply by $(n-1)!$ for a well-chosen large $n$, and trap the fractional part strictly between $0$ and $1$; the choice of $n$ needs primes $n$ for which a shifted value is an almost-prime of a prescribed shape, supplied in the interval $[x/2,x]$ by a version of Chen's theorem stated as the paper's Theorem 3. The source card friedlander_2007_irrationality_divisor_function_series holds the journal's PDF (the first paper link; the second is the journal's Zenodo deposit). The paper's Theorem 2, the irrationality of every case under the prime $k$-tuples conjecture, is the conditional claim on its own page. A note added in March 2007 records that Schlage-Puchta obtained the same two results independently, with a rather different sieve proof for $k=3$ (his 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: Integers, volume 7 (2007), article A31, received 8 December 2006, revised 20 March 2007, accepted 12 June 2007 and published 3 July 2007, as the paper's header prints; Integers is a refereed electronic journal. The site labels the problem OPEN, so its curator's remark crediting this paper and Schlage-Puchta 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.