Denominators of coefficients in Stirling's expansion for log(Γ(z))

Conjecture I: if n > 2, then a(A005382(n))/12 is prime, where A005382 is the sequence of primes p such that 2p-1 is also prime. Since A005382(1) = 2, A005382(2) = 3 and A005382(3) = 7, this says that a(p)/12 is prime for every prime p > 3 such that 2p-1 is also prime.

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-46969,
  title        = {Denominators of coefficients in Stirling's expansion for log(Γ(z))},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/oeis-46969}},
  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

Conjecture I: if $n > 2$ , then $\frac{a ( A005382 ( n ))}{12}$ is prime, where A005382 is the sequence of primes $p$ such that $2 p - 1$ is also prime. Since $A005382 (1) = 2$ , $A005382 (2) = 3$ and $A005382 (3) = 7$ , this says that $\frac{a ( p )}{12}$ is prime for every prime $p > 3$ such that $2 p - 1$ is also prime.

Lorenzo Sauras Altuzarra, Oct 13 2020

The $n$ -th term is the denominator of $\frac{B _{2 n}}{2 n ( 2 n - 1 )}$ where $B_{2 n}$ is the $2 n$ -th Bernoulli number.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.OEIS.«46969».

theorem conjecture1 (p : ℕ) (hp : p.Prime) (hp' : (2 * p - 1).Prime) (h3 : 3 < p) :
    (a p / 12).Prime

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

A046969

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.