slug: oeis-a116184-generalized-harmonic-thirty-seven-cube-progression bibkey: adamchuk2007a116184 doi: null url: https://oeis.org/A116184 triage: theorem motivation_gids:

• D5/S3/ArithSums/AdamchukGeneralizedHarmonicThirtySevenCubeProgression

Cubic divisibility of generalized harmonic numerators along Adamchuk’s progression

Problem

OEIS A116184, NAME (%N, verbatim):

Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ].

COMMENT (%C, verbatim):

Conjecture: All terms of the arithmetic progression 3+36k belong to a(n).

AUTHOR (%A, verbatim):

Alexander Adamchuk, Apr 08 2007

The first terms (%S) are:

3, 37, 39, 73, 75, 111, 147, 148, 183, 185, …

The literal claim is ∀ k ∈ ℕ, 37³ ∣ num(H(36, 3+36k)), where Rat.num is the reduced numerator. The same OEIS comment also discusses primes of the form 37m-1 and cases of divisibility by 37^4; neither assertion is claimed here.

Motivation

The comment singles out an infinite arithmetic progression inside the sequence. A universal proof settles every term of that progression rather than extending the finite list of verified exponents.

Gap

The surfaces recorded in preregistration issue #7581 and its probe were checked on 2026-09-13. The OEIS entry had six revisions: the conjecture was unchanged from revision 2 onward, and the later revisions changed only keywords, formatting, or links. OEIS Open arXiv:2608.11941, epoch-research/LeanOpenProblems, and google-deepmind/formal-conjectures did not contain this problem. Exact-phrase arXiv searches returned 0 results, and MathOverflow searches returned 0 results. OpenAlex returned HTTP 429, so its result is ASSUMED-UNVERIFIED.

The search seat reported that the case k=0 followed from Zhao Theorem 2.8 or Gy Theorem 3.2 together with the irregular pair (37,32). The probe could not identify those references, and arXiv:math/0303332v3 has no Theorem 2.8. That literature report is ASSUMED-UNVERIFIED and is not used. Wolstenholme-type congruences of generalized harmonic numbers are classical; no priority claim is made for the progression or its proof.

Route

Let L=36!, b_j=L/j, and N(n)=Sum_j b_j^n, so that H(n)=N(n)/L^n. In ZMod(37^3), Fermat gives 37 | b_j^36-1, hence (b_j^36-1)^3=0. Therefore C_k=Sum_j b_j^3*(b_j^36)^k satisfies C_(k+3)-3*C_(k+2)+3*C_(k+1)-C_k=0. Finite certificates establish C_0=C_1=C_2=0, and strong induction gives C_k=0 for every natural k.

For the numerator bridge, den(H(n)) | L^n, and L^n is coprime to 37. The identity num(H(n))*L^n=N(n)*den(H(n)) transfers divisibility by 37^3 from N(3+36k) to the reduced numerator.

Falsifier

One natural k for which 37^3 does not divide the reduced numerator of H(36,3+36k) would contradict the theorem.

Evidence

• Lean module: D5/S3/ArithSums/AdamchukGeneralizedHarmonicThirtySevenCubeProgression.lean.
• Main theorem: adamchuk_a116184.
• Axioms: std3 (propext, Classical.choice, Quot.sound).
• The orchestrator checked N(3+36k) = 0 (mod 37^3) for 0 <= k < 400 and exact rational values for k=0,1,2,3.
• The orchestrator found that {1 <= n < 340 : 37^3 | num(H(36,n))} equals the displayed %S terms in that range.
• With r_j=b_j^36, the orchestrator checked r_j = 1 (mod 37), (r_j-1)^3 = 0 (mod 37^3), and C_0=C_1=C_2=0.
• The independent probe recorded the same modular, rational, sequence, and certificate readings before proving the universal statement.

Triage

theorem. The formal result proves the full arithmetic progression for every natural k.

ASSUMED-UNVERIFIED

OpenAlex completeness was not checked because the service returned HTTP 429. The search seat’s Zhao/Gy citation could not be identified and is not relied upon. All search results are bounded to the named surfaces and date; they do not establish exhaustive literature coverage or publication priority.