Hanna’s Cubed Sigma-Product Exponential

Abstract

The cube has the characteristic-three coefficient pattern at every index.

Definition 1.1 (The A382125 congruences).

$$\mathrm{claim}\left(\right)⟺((\forall n:\mathbb{N},\exists u:\mathbb{Z},\exists v:\mathbb{Z},(\mathrm{coeff}\left(3\cdot n,B\left(\right)\right)=(u:\mathbb{Q}))\wedge ((\mathrm{coeff}\left(n,A\left(\right)\right)=(v:\mathbb{Q}))\wedge (u\equiv v(\mathrm{mod}3))))\wedge (\forall n:\mathbb{N},\exists u:\mathbb{Z},\exists v:\mathbb{Z},(\mathrm{coeff}\left(3\cdot n+1,B\left(\right)\right)=(u:\mathbb{Q}))\wedge ((\mathrm{coeff}\left(3\cdot n+2,B\left(\right)\right)=(v:\mathbb{Q}))\wedge ((u\equiv 0(\mathrm{mod}3))\wedge (v\equiv 0(\mathrm{mod}3))))))$$

Formalization. D5/S3/Arith/Robin/HannaSigmaProductExpModThreeCube.claim (✓ std3).

Citation. Paul D. Hanna (2025). OEIS A382124 and A382125, sigma-product exponential series. URL: https://oeis.org/A382124.

Commentary.

Conjectures: a(3n) == A382124(n) (mod 3) for n >= 0; a(3n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0. OEIS A382125, COMMENT (unpaginated). A and B are the rational generating functions defined in HannaSigmaProductExpModThree; all coefficient witnesses are integers.

Theorem 1.2 (All three coefficient classes).

$$(\forall n:\mathbb{N},\exists u:\mathbb{Z},\exists v:\mathbb{Z},(\mathrm{coeff}\left(3\cdot n,B\left(\right)\right)=(u:\mathbb{Q}))\wedge ((\mathrm{coeff}\left(n,A\left(\right)\right)=(v:\mathbb{Q}))\wedge (u\equiv v(\mathrm{mod}3))))\wedge (\forall n:\mathbb{N},\exists u:\mathbb{Z},\exists v:\mathbb{Z},(\mathrm{coeff}\left(3\cdot n+1,B\left(\right)\right)=(u:\mathbb{Q}))\wedge ((\mathrm{coeff}\left(3\cdot n+2,B\left(\right)\right)=(v:\mathbb{Q}))\wedge ((u\equiv 0(\mathrm{mod}3))\wedge (v\equiv 0(\mathrm{mod}3)))))$$

Proof. Machine-checked in Lean as D5/S3/Arith/Robin/HannaSigmaProductExpModThreeCube.result (✓ std3). ∎

Resolves. Problems/oeis-a382125-hanna-sigma-exponential-mod-three-cube (proved) by D5/S3/Arith/Robin/HannaSigmaProductExpModThreeCube.result.

Source. Repository-derived.

Acknowledgement. Paul D. Hanna (2025). OEIS A382124 and A382125, sigma-product exponential series. URL: https://oeis.org/A382124.

Commentary.

The first congruence is the symmetric A382124 congruence. The other two follow from the characteristic-three coefficient pattern of the integer lift cubed.

References

• Truth anchor: D5/S3/Arith/Robin/HannaSigmaProductExpModThreeCube.claim
• Truth anchor: D5/S3/Arith/Robin/HannaSigmaProductExpModThreeCube.result
• Dependency: D5/S3/Arith/Robin/HannaSigmaProductExpModThree