Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is\[\sum \frac{\sigma_k(n)}{n!}\]irrational?
This is known now for $1\leq k\leq 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erd\H{o}s \cite{Er52}. The case $k=3$ was proved independently by Schlage-Puchta \cite{ScPu06} and Friedlander, Luca, and Stoiciu \cite{FLC07}. The case $k=4$ was proved by Pratt \cite{Pr22}.
It is known that this sum is irrational for all $k\geq 1$ conditional on either Schinzel's conjecture (Schlage-Puchta \cite{ScPu06}) or the prime tuples conjecture (Friedlander, Luca, and Stoiciu \cite{FLC07}).
This is discussed in problem B14 of Guy's collection \cite{Gu04}.
References
[Er52] Erd\H{o}s, P., Problem 4493. Amer. Math. Monthly (1952), 557-558.
[FLC07] Friedlander, J. B. and Luca, F. and Stoiciu, M., On the irrationality of a divisor function series. Integers (2007).
[Gu04] Guy, Richard K., Unsolved problems in number theory. (2004), xviii+437.
[Pr22] Pratt, K., The irrationality of a divisor function series of Erd\H{o}s and Kac. arXiv:2209.11124 (2022).
[ScPu06] Schlage-Puchta, J. C., The irrationality of a number theoretical series. Ramanujan J. (2006), 455-460.
