A Proof of Bala's Congruence Conjectures for A158690

Abstract

Let $a(n)$ be the sequence A158690 in the On-Line Encyclopedia of Integer Sequences (OEIS), defined by the exponential generating function ${\sum }_{n\ge 0}a(n)t^n/n!=1+{\sum }_{m\ge 1}{\prod }_{j=1}^m(1-e^{-(2j-1)t})$. We prove two congruence conjectures of Peter Bala. The first states that, for every integer $k\ge 1$, the sequence $a(n)$ modulo $k$ is eventually periodic with period dividing $\phi (k)$. We prove the stronger statement that the Carmichael function $\lambda (k)$ is an eventual period. The second conjecture asserts the shifted Gauss congruences $a(n{p}^r+i)\equiv a(n{p}^{r-1}+i)(\mathrm{mod}{p}^r)$ for every $i\ge 0$, every prime $p$, and all $n,r\ge 1$. Both results follow from a general theorem for exponential generating functions of the form $G(e^t-1)$ with $G\in \mathbb{Z}[[y]]$, together with the standard power-sum formula for Stirling numbers of the second kind.

arXiv comment7 pages. Includes a Lean 4/Mathlib formalization