Numerator of 1/det(M) for M[i,j] = 1/lcm(i,j)

"Conjecture: 1/det(M) is an integer only for n: 1 to 34, 36 and 38." - _Robert G. Wilson v_, Aug 02 2015

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-60841,
  title        = {Numerator of 1/det(M) for M[i,j] = 1/lcm(i,j)},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-60841}},
  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

"Conjecture: $1/ det (M)$ is an integer only for n: 1 to 34, 36 and 38." - _Robert G. Wilson v_, Aug 02 2015

Numerator of $1/ det (M)$ where $M$ is the $n \times n$ matrix with $M [i, j] = 1/ lcm (i, j)$ .

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«60841».

theorem conjecture1 :
    ∀ n : ℕ, 1 ≤ n → (((lcmMatrix n).det)⁻¹.den = 1 ↔ n ∈ integerDetN)

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

A060841
T. Adamczewski, "OEIS Open: How many conjectures can language models turn into theorems?", arXiv:2608.11941 [cs.AI], 2026.

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.