2016/05/31 by Paolo Leonetti, P. Leonetti, Carlo Sanna +1 · 7 citations
Computer Science · Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Base (topology) #Conjecture #Constant (computer programming) #Existential quantification #Stirling numbers of the second kind #Valuation (finance) #math.CO #math.NT #msc:11A51 #msc:11B50 #msc:11B73 #semigroups and automata theory
paper · pdf · doi:10.1007/s10474-016-0680-4
published in Acta Mathematica Academiae Scientiarum Hungaricae 151(1), 217-231 (Springer Nature) · 12 pages, 3 figures
arxiv created 2016/06/04 · openalex created_date 2016/06/24 · openalex publication_date 2016/12/19 · arxiv updated 2017/08/29 · openalex updated_date 2026/08/05
For all integers n ≥ k ≥ 1, define H(n,k) := ∑ 1 / (i1 ⋯ ik), where the sum is extended over all positive integers i1 < ⋯ < ik ≤ 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 Erdős-Niven and Chen-Tang, we study the p-adic valuation of H(n,k). In particular, for any prime number p, integer k ≥ 2, and x ≥ (k-1)p, we prove that νp(H(n,k)) < -(k - 1)(logp(n/(k - 1)) - 1) for all positive integers n ∈ [(k-1)p, 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.