---

Claim. Theorem 2 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 the prime $k$-tuples conjecture implies that

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

is irrational, stated for every positive integer $k$ and proved in the paper's Section 3 for $k\ge4$, the cases $k\le3$ being covered by its Theorem 1 and the short proofs for $k\le2$ in its introduction; the abstract phrases the result as holding for $k\ge4$. This would answer Problem 252 yes for every $k\ge1$. The claim is conditional: the hypothesis is the paper's Conjecture 1, Dickson's form of the prime $k$-tuples conjecture for linear polynomials, that for $k\ge2$, integers $a_i>0$ and $b_i$ such that no prime divides $\prod_i(a_in+b_i)$ for every $n$, infinitely many positive $n$ make every $a_in+b_i$ prime; it is unproven, so this page derives nothing for the problem's standing. 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 unconditional Theorem 1, the case $k=3$, is the partial claim on its own page; Schlage-Puchta proved a parallel conditional theorem under Schinzel's Hypothesis H (his conditional page), as the paper's note added in March 2007 records. formal-conjectures tags its variant erdos_252.variants.prime_tuples, which takes the conjecture as an explicit hypothesis for $k\ge4$, research solved, citing this paper.

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 that the sum is irrational for every $k$ under Dickson's conjecture by this paper 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.