Erdős Problem #1060 — open problem in number theory

Erdős Problem #1060

The conjecture is about the function f(n) which counts the number of solutions to kσ(k)=n, where σ(k) is the sum of divisors of k. The first bound is that f(n) grows slower than any power of n^(1/loglog n). The second bound is that f(n) is at most a power of log n.

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.

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Cite

@misc{cairn-erdos-1060,
  title        = {Erdős Problem #1060},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-1060}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

Verified

Disputed

Refuted

On the literature board

The problem

The question

erdos_1060.parts.i. The conjecture is about the function $f (n)$ which counts the number of solutions to $k σ (k) = n$ , where $σ (k)$ is the sum of divisors of $k$ . The first bound is that $f (n)$ grows slower than any power of $n^{(} \frac{1}{l o g l o g n})$ . The second bound is that $f (n)$ is at most a power of $lo g n$ .

erdos_1060.parts.ii. Part (ii) of Erdős Problem 1060: bound on the number of $k \leq n$ with $k σ_{1} (k) = n$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1060» (2 statements).

theorem erdos_1060.parts.i :
    ∃ h : ℕ → ℝ,
      h =o[atTop] (fun n ↦ 1 / log (log n)) ∧ ∀ᶠ n in atTop, #{k ≤ n | k * σ 1 k = n} ≤ (n : ℝ) ^ h n

theorem erdos_1060.parts.ii :
    ∃ (C : ℝ), (fun n ↦ (#{k ≤ n | k * σ 1 k = n} : ℝ)) =O[atTop]
      (fun n ↦ log n ^ C)

What counts as progress

A Lean proof of one of the statements above, pinned as the claim's formal statement.
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/1060. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/1060

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.