bibkey: ianakiev2020a336687
authors: Ivan N. Ianakiev; Bernard Schott
year: 2020
title: OEIS A336687, divisor compositions and a three-solution equality conjecture
doi: null
url: https://oeis.org/A336687
claim: Ivan N. Ianakiev conjectured that tau(sigma(m))*sigma(tau(m)) = m has only the positive solutions 1, 468, and 3240, and reported verification through 10^9.
strata_touched:
  - D5/S3/Factorization/TauSigmaPowerBounds
  - D5/S3/Factorization/TauSigmaSolutionBound
license: citation-only
triage: anchor

OEIS A336687

The entry itself, created by Bernard Schott on 2020-07-31, concerns numbers m for which tau(sigma(m)) and sigma(tau(m)) both divide m. The equality conjecture is a separate comment by Ivan N. Ianakiev dated 2020-08-06. His given name is Ivan, not Kival. The internal entry was read directly on 2026-09-09 (Singapore time), at revision #24 Aug 06 2020 05:05:55.

The comment states:

Conjecture: The only m such that m = tau(sigma(m))sigma(tau(m)) are in {1,468,3240}. Verified for m up to 110^9. - Ivan N. Ianakiev, Aug 06 2020

The repository proves the two uniform estimates tau(n)^4 <= 9^4n and sigma(n)^4 <= 3n^5, and derives m < 3^13 for every positive equality solution. These bounds are repository-derived from the preregistered prime-power argument; they are not attributed to Ianakiev or Schott. The finite classification is an explicit unproved hypothesis in the companion equivalence theorem.

This implementation does not prove the A336687 conjecture. The OEIS page still labels the equality statement a conjecture; no proof was found in the inspected entry or the identifier-based GitHub Lean search. That limited search does not establish that no proof exists elsewhere.

Verified locator

• https://oeis.org/A336687
• https://oeis.org/A336687/internal
• https://oeis.org/search?q=id:A336687&fmt=text: the name line, the offset, the author line, the term list, and Ianakiev’s 2020-08-06 comment carrying the equality conjecture and its verification range.

Scope of the formal work

The two uniform estimates and the bound m < 3^13 are proved without hypotheses. The three-solution equivalence carries the finite classification over the window below 3^13 as an explicit hypothesis, and that hypothesis is not discharged here.

ASSUMED-UNVERIFIED

First-publication priority is not established, and the search for a later proof was limited to the entry itself and an identifier-based code search. Whether the volume’s prime-power maxima match the ones used in the Lean proof was checked by reading, not by machine.