On the p-adic valuation of Stirling numbers of the first kind

For all integers $n \geq k \geq 1$, define $H(n,k) := \sum 1 / (i_1 \cdots i_k)$, where the sum is extended over all positive integers $i_1 < \cdots < i_k \leq n$. These quantities are closely related to the Stirling numbers of the first kind by the identity ${H(n,k) = s(n + 1, k + 1) / n!}$. Motivated by the works of Erd\H{o}s--Niven and Chen--Tang, we study the $p$-adic valuation of $H(n,k)$. Lengyel proved that $\nu_p(H(n,k))>-k\log_p n+O_k(1)$ and we conjecture that there exists a positive constant $c=c(p,k)$ such that $\nu_P(H(n,k))<-c\log n$ for all large $n$. In this respect, we prove the conjecture in the affirmative for all $n \leq x$ whose base $p$ representations start with the base $p$ representation of $k - 1$, but at most $3 x^{0.835}$ exceptions. We also generalize a result of Lengyel by giving a description of $\nu_2(H(n,2))$ in terms of an infinite binary sequence.