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In recent years, Lengyel, Komatsu and Young, Leonetti and Sanna, Adelberg, Hong and Qiu made some progress in the study of the $p$-adic valuations of $s(n,k)$. In this paper, by using Washington's congruence on the generalized harmonic number and the $n$-th Bernoulli number $B_n$ and the properties of $m$-th Stirling numbers of the first kind obtained recently by the authors, we arrive at an exact ex"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1908.05594","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-08-15T15:44:10Z","cross_cats_sorted":[],"title_canon_sha256":"47bb6ed6f5584c8eac90cad9cf876b0fbc3e10047331e159094ac286a9511bcd","abstract_canon_sha256":"15be1acfc166d6a5e7b9a699683f5c9818bb637b295bbd3372edf7b94851378a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T00:44:39.658184Z","signature_b64":"a12x0uUMab+vhKQ77eAvcIl167Gg2Vg9d2Pd3keT++pI9mr0MfYLfs85rGpud4T1dJqM0ubAUuZxiMD4pVcrAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d54a3c9bea63c8577838d3847481eb467788a215ece6110a8e6eb02fbb065c0b","last_reissued_at":"2026-07-05T00:44:39.657718Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T00:44:39.657718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the $p$-adic properties of Stirling numbers of the first kind","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Min Qiu, Shaofang Hong","submitted_at":"2019-08-15T15:44:10Z","abstract_excerpt":"Let $n, k$ and $a$ be positive integers. 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