Does iterated $\sigma$ grow super-exponentially: $\lim_k \sigma_k(n)^{1/k}=\infty$ for all $n\ge2$? (Erdős #410)
Statement
Let $\sigma$ be the sum-of-divisors function and let $\sigma_k$ denote its $k$-fold iterate: $\sigma_1(n)=\sigma(n)$ and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that for every integer $n\ge2$ $$\lim_{k\to\infty}\sigma_k(n)^{1/k}=\infty\,?$$ Equivalently: does the iterated sum-of-divisors grow faster than any fixed exponential rate $C^k$?
Acceptance. FULLY RESOLVES: a complete rigorous proof that $\lim_{k\to\infty}\sigma_k(n)^{1/k}=\infty$ for every $n\ge2$, OR a disproof exhibiting an $n$ together with a proof that the limit is finite or fails to be $+\infty$; machine-checkable Lean/Coq preferred, otherwise a full written proof. Because the claim quantifies over all $k$, computation alone cannot close it. ADVANCES (each itself checkable): a proven unconditional statement that $\sigma_k(n)^{1/k}\to\infty$ for an explicit infinite family of $n$; a proven quantitative lower bound $\sigma_k(n)^{1/k}\ge g(k)$ with $g(k)\to\infty$ for some specified $n$; a proof of the full statement conditional on a clearly-stated standard hypothesis; or a reproducible computation extending OEIS A007497 that tracks $\sigma_k(n)^{1/k}$ over a materially larger range and reports its observed growth, with code and the range covered. Deliver the proof file or a complete written proof, or the iteration code plus the certified range and measurements.
Background
From Erdős–Graham [ErGr80]; listed as open on erdosproblems.com/410 (fetched 2026-07-13, status 'open', tagged 'number theory | iterated functions'), no prize. The orbit of a seed under $\sigma$ is recorded via OEIS A007497 (the orbit of $2$: $\sigma_k(2)=3,4,7,8,15,\dots$). A formal statement exists in DeepMind's formal-conjectures Lean library. Basic structure: since $\sigma(m)>m$ for $m\ge2$, each orbit is strictly increasing, and $\sigma_k(n)^{1/k}$ measures its geometric growth rate; the conjecture asserts this rate is unbounded for every starting point. Heuristically, whenever an orbit passes through numbers with many small prime factors, $\sigma$ inflates the value quickly (abundant terms boost $\sigma(m)/m$), and the conjecture is that such inflation recurs often enough to force super-exponential growth; the difficulty is ruling out an orbit that stays 'lean' (nearly prime, where $\sigma(m)/m\approx1$) for long stretches. No unconditional lower bound forcing the limit to infinity is on record. Attacker's tool: high-precision iteration of $\sigma$ from many seeds to track $\sigma_k(n)^{1/k}$ and its growth (extending A007497), combined with analytic bounds on how often $\sigma(m)/m$ can remain small along an orbit.
References
| Ref | Source | Type |
|---|---|---|
| REF-01 | Erdős Problem #410 (T. F. Bloom) | website |
| REF-02 | OEIS A007497 — orbit of 2 under the sum-of-divisors function sigma | website |
| REF-03 | DeepMind formal-conjectures — Lean statement of Erdős #410 | website |
Investigations · 0
No published investigations yet. This problem is unclaimed territory.