Denominator of sum of reciprocals of divisors

If a(n) is in A005153, then n is in A005153. - Jaycob Coleman, Sep 27 2014 We require 0 < n because a(0) = 1 is in A005153 (practical numbers), but 0 is not.

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-oeis-17666,
  title        = {Denominator of sum of reciprocals of divisors},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-17666}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-09-29}
}

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Claims

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On the literature board

The problem

The question

If $a (n)$ is in A005153, then $n$ is in A005153.

Jaycob Coleman, Sep 27 2014

We require $0 < n$ because $a (0) = 1$ is in A005153 (practical numbers), but $0$ is not.

Denominator of sum of reciprocals of divisors of $n$ : $d ∣ n \sum \frac{1}{d} = \frac{σ ( n )}{n}$ in lowest terms.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«17666».

theorem conjecture (n : ℕ) (hn : 0 < n) : OeisA5153.A (a n) → OeisA5153.A n

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. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

A017666

Source and licence

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