OPEN
This is open, and cannot be resolved with a finite computation.
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].
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].
View the LaTeX source
This page was last edited 02 December 2025.
Additional thanks to: Quanyu Tang
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #252, https://www.erdosproblems.com/252, accessed 2026-01-16
