Is $\sum_n \sigma_k(n)/n!$ irrational for every $k\ge1$? (Erdős #252)
Statement
For an integer $k\ge1$ let $\sigma_k(n)=\sum_{d\mid n} d^k$ be the sum of the $k$th powers of the divisors of $n$ (so $\sigma_1=\sigma$ is the ordinary sum-of-divisors function). Is the real number $$\sum_{n=1}^{\infty}\frac{\sigma_k(n)}{n!}$$ irrational for every $k\ge1$?
Acceptance. FULLY RESOLVES: a complete rigorous proof that $\sum_n \sigma_k(n)/n!$ is irrational for every $k\ge1$ unconditionally — machine-checkable Lean/Coq preferred, otherwise a full written proof with all steps. ADVANCES (each itself checkable): an unconditional irrationality proof for a specific new value of $k$ strictly beyond the best stated in the background — the background records unconditional proofs only up to $k=4$ (Erdős; Schlage-Puchta; Friedlander–Luca–Stoiciu; Pratt), so $k=5$ or higher qualifies; OR an irrationality proof for all $k$ under a clearly-stated hypothesis that is weaker than or different from Schinzel's Hypothesis H and Dickson's conjecture; OR a new unconditional structural reduction that provably lowers the prime-tuple input required. A proof must accompany any claim; numerical evidence alone is insufficient. Deliver the Lean proof file or a complete written proof, naming the exact $k$ resolved and any hypotheses used.
Background
Posed by Erdős, recorded in Erdős–Graham [ErGr80, p.62] and Erdős [Er88c, p.102], and discussed as problem B14 in Guy's Unsolved Problems in Number Theory [Gu04]; listed as open on erdosproblems.com/252 (fetched 2026-07-13, status 'open', tagged 'number theory | irrationality'), with no prize. Known progress: the statement is proven for $1\le k\le 4$. The cases $k=1,2$ are reasonably straightforward, as observed by Erdős [Er52]; $k=3$ was proved independently by Schlage-Puchta [ScPu06] and by Friedlander, Luca, and Stoiciu [FLC07]; and $k=4$ was proved by Pratt [Pr22]. Moreover the full statement (all $k\ge1$) is known to follow from standard prime-tuple hypotheses: Schlage-Puchta [ScPu06] deduces it from Schinzel's Hypothesis H, and Friedlander–Luca–Stoiciu [FLC07] from Dickson's conjecture. So the frontier is: the first unconditional open case is $k=5$, and an unconditional resolution for all $k$ appears to require new input beyond current prime-tuple technology. Related OEIS sequences are A227988 and A227989, and a formal statement exists in DeepMind's formal-conjectures Lean library. Irrationality is not a finite check; the attacker's tool is the analytic irrationality method used for $k\le4$ (controlling, for the series tail modulo $q$, the existence of $n$ with prescribed factorisation via sieve / prime-tuple inputs), pushed either to $k=5$ unconditionally or to all $k$ under a stated hypothesis — possibly delivered as a Lean formalisation.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #252 (T. F. Bloom) | website |
| REF-02 | OEIS A227988 — related to irrationality of sum sigma_k(n)/n! | website |
| REF-03 | OEIS A227989 — related to irrationality of sum sigma_k(n)/n! | website |
| REF-04 | DeepMind formal-conjectures — Lean statement of Erdős #252 | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.