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Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbersby $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They areintegers. We introduce and study a variant of Kummer's notion of regularity ofprimes. We say that an odd prime $p$ is $\ell$-Genocchi irregular if it dividesat 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 ofArtin's primitive root conjecture, we give asymptotic estimates for the numberof $\ell$-Genocchi irregular primes in a prescribed arithmetic progression incase $\ell$ is odd. The case $\ell=2$ was already dealt with by Hu, Kim, Moreeand 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 $p\leqx$ for which there exist modulo $p$ Ramanujan-style congruences between theFourier coefficients of an Eisenstein series and some cusp form of prime level$\ell$.

Prime divisors of $\ell$-Genocchi numbers and the ubiquity of Ramanujan-style congruences of level $\ell$