Problem 252

Erdős Problem #252: declared status 'open'. Formalized: yes. Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$. Is$$\sum \frac{\sigma_k(n)}{n!}$$irrational? Current best: This is known now for $1\leq k\leq 4$. It is known that this sum is irrational for all $k\geq 1$ conditional on either Schinzel's conjecture (Schlage-Puchta [ScPu06]) or the prime tuples conjecture (Friedlander, Luca, and Stoiciu [FLC07]). Prize: no. OEIS: A227988, A227989. Tags: irrationality, number theory.