Erdős Problem #410 — open problem in number theory

Erdős Problem #410

Let σ_1(n) = σ(n), the sum of divisors function, and σ_k(n) = σ(σ_k-1(n)). Is it true that lim_k → ∞ σ_k(n)^frac 1 k = ∞?

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim. A Lean proof is checked against the statement below by the Lean kernel; a curator confirms before the problem counts as resolved.

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Cite

@misc{cairn-erdos-410,
  title        = {Erdős Problem #410},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-410}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-28}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

Let $σ_{1} (n) = σ (n)$ , the sum of divisors function, and $σ_{k} (n) = σ (σ_{k - 1} (n))$ .

Is it true that $lim_{k \to \infty} σ_{k} (n)^{\frac{1}{k}} = \infty$ ?

This is problem (iii) from Erdos, Granville, Pomerance, Spiro "On the normal behavior of the iterates of some arithmetical functions" (page 169 of the book "Analytic Number Theory", 1990).

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«410». answer(sorry) marks a yes/no question: a proof of the statement on the right of ↔, or of its negation, answers it.

theorem erdos_410 : answer(sorry) ↔ ∀ n > 1,
    Tendsto (fun k : ℕ ↦ ((sigma 1)^[k] n : ℝ) ^ (1 / (k : ℝ))) atTop atTop

What counts as progress

A Lean proof of the pinned statement (or of its negation, for a yes/no question) — checked by the Lean kernel against the upstream statement; a curator confirms before the problem is marked resolved.
Partial results: special cases, weaker bounds, reductions — as verified claims.
Computations and numerical evidence with published code (reproducible).
Literature: the problem may have been solved or partly solved already — check erdosproblems.com/410. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/410

Source and licence

Imported from Formal Conjectures (Erdős problems), commit e6d1743831c2. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.