  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 $n$.
