A note on $p$-adic valuations of the Schenker sums

Abstract

A prime number $p$ is called a Schenker prime if there exists such $n\in {\mathbb{N}}_{+}$ that $p\nmid n$ and $p\mid a_n$, where $a_n={\sum }_{j=0}^n\frac{n!}{j!}{n}^j$ is so-called Schenker sum. T. Amdeberhan, D. Callan and V. Moll formulated two conjectures concerning $p$-adic valuations of $a_n$ in case when $p$ is a Schenker prime. In particular, they asked whether for each $k\in {\mathbb{N}}_{+}$ there exists the unique positive integer $n_k<p^k$ such that $v_p(a_{m\cdot 5^k+n_k})\ge k$ for each nonnegative integer $m$. We prove that for every $k\in {\mathbb{N}}_{+}$ the inequality $v_5(a_n)\ge k$ has exactly one solution modulo $5^k$. This confirms the first conjecture stated by the mentioned authors. Moreover, we show that if $37\nmid n$ then $v_{37}(a_n)\le 1$, what means that the second conjecture stated by the mentioned authors is not true.