Exact p-adic valuations of Stirling numbers of the first kind

For any positive integer k and prime p, we define an explicit set Ak,p of positive integers n, having positive upper and lower density, on which the p-adic valuation of the Stirling number of the first kind s(n+1,k+1) is a nondecreasing function of n and is given exactly by a simple formula. This is motivated by, and extends, some recent results of Lengyel.