Arithmetic properties of the Genocchi numbers and their generalization

TL;DR

This paper investigates new arithmetic properties of generalized Genocchi numbers, including divisibility features, and extends these findings to classical Genocchi numbers, revealing their relation to the odd part of even integers.

Contribution

It establishes novel divisibility properties of generalized Genocchi numbers and derives implications for the classical Genocchi numbers, enhancing understanding of their arithmetic structure.

Findings

Genocchi numbers are divisible by the odd part of even integers

New divisibility properties for generalized Genocchi numbers

Extensions of properties to classical Genocchi numbers

Abstract

This note is devoted to establish some new arithmetic properties of the generalized Genocchi numbers $G_{n , a}$ ($n \in \mathbb{N}$, $a \geq 2$). The resulting properties for the usual Genocchi numbers $G_n = G_{n , 2}$ are then derived. We show for example that for any even positive integer $n$, the Genocchi number $G_n$ is a multiple of the odd part of $n$.