  Let $B_n$ ($n = 0, 1, 2, ...$) denote the usual $n$-th Bernoulli number. Let
$l$ be a positive even integer where $l=12$ or $l \geq 16$. It is well known
that the numerator of the reduced quotient $|B_l/l|$ is a product of powers of
irregular primes. Let $(p,l)$ be an irregular pair with $B_l/l \not\equiv
B_{l+p-1}/(l+p-1) \modp{p^2}$. We show that for every $r \geq 1$ the congruence
$B_{m_r}/m_r \equiv 0 \modp{p^r}$ has a unique solution $m_r$ where $m_r \equiv
l \modp{p-1}$ and $l \leq m_r < (p-1)p^{r-1}$. The sequence $(m_r)_{r \geq 1}$
defines a $p$-adic integer $\chi_{(p, l)}$ which is a zero of a certain
$p$-adic zeta function $\zeta_{p, l}$ originally defined by T. Kubota and H. W.
Leopoldt. We show some properties of these functions and give some
applications. Subsequently we give several computations of the (truncated)
$p$-adic expansion of $\chi_{(p, l)}$ for irregular pairs $(p,l)$ with $p$
below 1000.
