Adamchuk’s A116184 progression

Abstract

Every exponent in Adamchuk’s progression gives a generalized harmonic numerator divisible by 37 cubed.

Definition 1.1 (The generalized harmonic sum).

$$\forall n\in \mathbb{N},H\left(n\right)=\sum _{j\in Finset.Icc\left(1,36\right)}\frac{(1:\mathbb{Q})}{(j:\mathbb{Q}{)}^n}$$

Formalization. D5/S3/ArithSums/AdamchukGeneralizedHarmonicThirtySevenCubeProgression.H (✓ std3).

Citation. Alexander Adamchuk (2007). OEIS A116184, Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ]. URL: https://oeis.org/A116184.

Commentary.

This is the generalized harmonic number H(36, n) of OEIS A116184; Rat.num below is the reduced numerator.

Theorem 1.2 (Cubic divisibility along the progression).

$$\forall k\in \mathbb{N},(37^3:\mathbb{Z})\mid num\left(H\left(3+36\cdot k\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/ArithSums/AdamchukGeneralizedHarmonicThirtySevenCubeProgression.adamchuk_a116184 (✓ std3). ∎

Resolves. Problems/oeis-a116184-generalized-harmonic-thirty-seven-cube-progression (proved) by D5/S3/ArithSums/AdamchukGeneralizedHarmonicThirtySevenCubeProgression.adamchuk_a116184.

Source. Repository-derived.

Commentary.

Cubic nilpotence gives a third-order recurrence. Three initial certificates and induction make every recurrence value vanish, and the numerator bridge transfers that divisibility to the reduced numerator of the harmonic sum.

References

• Truth anchor: D5/S3/ArithSums/AdamchukGeneralizedHarmonicThirtySevenCubeProgression.H
• Truth anchor: D5/S3/ArithSums/AdamchukGeneralizedHarmonicThirtySevenCubeProgression.adamchuk_a116184