Problem #252: Let $k\geq 1$ and $\sigma_k(n)=\sum_{d\mid n}d^k$

---

Progress

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].

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]).

This is discussed in problem B14 of Guy's collection [Gu04].