OPEN

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 Erdős

[Er52]. The case $k=3$ was proved independently by Schlage-Puchta

[ScPu06] and Friedlander, Luca, and Stoiciu

[FLC07]. The case $k=4$ was proved by Pratt

[Pr22].