The integrality of the Genocchi numbers obtained through a new identity and other results

Abstract

In this note, we investigate some properties of the integer sequence of general term a_n := \sum_{k = 0}^{n - 1} k! (n - k - 1)! (\forall n \geq 1) to derive a new identity of the Genocchi numbers G_n (n \in \mathbb{N}), which immediately shows that G_n \in \mathbb{Z} for any n \in \mathbb{N}. In another direction, we obtain nontrivial lower bounds for the 2-adic valuations of the rational numbers \sum_{k = 1}^{n} \frac{2^k}{k}.