  In this paper, we introduce and study a variant of Kummer's notion of
(ir)regularity of primes which we call G-irregularity. It is based on Genocchi
numbers $G_n$, rather than Bernoulli number $B_n.$ We say that an odd prime $p$
is G-irregular if it divides at least one of the integers $G_2,G_4,\ldots,
G_{p-3}$, and G-regular otherwise. We show that, as in Kummer's case,
G-irregularity is related to the divisibility of some class number.
Furthermore, we obtain some results on the distribution of G-irregular primes.
In particular, we show that each primitive residue class contains infinitely
many G-irregular primes and establish non-trivial lower bounds for their number
up to a given bound $x$ as $x$ tends to infinity. As a by-product, we obtain
some results on the distribution of primes in arithmetic progressions with a
prescribed near-primitive root.
