A note on $p$-adic valuations of Schenker sums
Volume 140 / 2015
Abstract
A prime number $p$ is called a Schenker prime if there exists $n\in\mathbb{N}_+$ such that $p\nmid n$ and $p\,|\, a_n$, where $a_n = \sum_{j=0}^{n}(n!/j!)n^j$ is a so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning $p$-adic valuations of $a_n$ when $p$ is a Schenker prime. In particular, they conjectured that for each $k\in\mathbb{N}_+$ there exists a unique positive integer $n_k<5^k$ such that $v_5(a_{m\cdot 5^k + n_k})\geq k$ for each nonnegative integer $m$. We prove that for every $k\in\mathbb{N}_+$ the inequality $v_5(a_n)\geq k$ has exactly one solution modulo $5^k$. This confirms the above conjecture. Moreover, we show that if $37\nmid n$ then $v_{37}(a_n)\leq 1$, which disproves the other conjecture of the above mentioned authors.
