  Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers
by $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They are
integers. We introduce and study a variant of Kummer's notion of regularity of
primes. We say that an odd prime $p$ is $\ell$-Genocchi irregular if it divides
at least one of the $\ell$-Genocchi numbers $G_2,G_4,\ldots, G_{p-3}$, and
$\ell$-regular otherwise. With the help of techniques used in the study of
Artin's primitive root conjecture, we give asymptotic estimates for the number
of $\ell$-Genocchi irregular primes in a prescribed arithmetic progression in
case $\ell$ is odd. The case $\ell=2$ was already dealt with by Hu, Kim, Moree
and Sha (2019).
  Using similar methods we study the prime factors of $(1-\ell^n)B_{2n}/2n$ and
$(1+\ell^n)B_{2n}/2n$. This allows us to estimate the number of primes $p\leq
x$ for which there exist modulo $p$ Ramanujan-style congruences between the
Fourier coefficients of an Eisenstein series and some cusp form of prime level
$\ell$.
