{"paper":{"title":"On the p-adic valuation of Stirling numbers of the first kind","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Carlo Sanna, Paolo Leonetti","submitted_at":"2016-05-24T12:43:41Z","abstract_excerpt":"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 $H(n,k)$. In particular, for any prime number $p$, integer $k \\geq 2$, and $x \\geq (k-1)p$, we prove that $\\nu_p(H(n,k)) < -(k - 1)(\\log_p(n/(k - 1)) - 1)$ for all positive integers $n \\in [(k-1)p, x]$ whose base $p$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1605.07424","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}