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

For all integers n≥k≥1, define H(n,k):=∑1/(i1⋯ik), where the sum is extended over all positive integers i1<⋯−klogpn+Ok(1) and we conjecture that there exists a positive constant c = c(p, k) such that νp(H(n,k))<−clogn for all large n. In this respect, we prove the conjecture in the affirmative for all n≤x whose base p representations start with the base p representation of k − 1, but at most 3x0.835 exceptions. We also generalize a result of Lengyel by giving a description of ν2(H(n,2)) in terms of an infinite binary sequence.