Let $p = 43, 67$ or $163$ (three primes such that $h(\mathbb{Q}(\sqrt{-p})) = 1$) and consider $k = (p+1)/2$. I'm interested in computing the $p$-adic valuation of the expression \begin{equation} 1+\frac{l}{B_l}, \end{equation} where $l = k + m(p-1),~m\geq 0$. Note that, by the Kummer congruences and the fact that $h(-p) \equiv -2B_k~(\mathrm{mod}~p)$, we have \begin{equation} v_p(1+l/B_l)\geq 1. \end{equation} In fact, I have observed by numerical computations that this inequality is an equality for many value of $m$. I have also observed that if $m$ is such that $2m+1 = dp$ for an integer $d$, then the valuation of $1+l/B_l$ seems to be equal to $2$.
These observations seem very mysterious for me, so I was wondering if anybody had an idea to explain this? Thanks !