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We prove that $\\gcd(a_n,a_m)=a_{\\gcd(n,m)}$ for all $n,m\\geqslant1$ if and only if $$a_n=\\prod\\limits_{d\\mid n} c_d\\quad\\mbox{for} \\ n\\geqslant1, $$ where $c_1=a_1$ and $c_n=\\mbox{lcm}(a_1,a_2,\\dots,a_n)/\\mbox{lcm}(a_1,a_2,\\dots,a_{n-1})$ for $n\\geqslant2$. All equalities with gcd and lcm are determined up to units of $R$."},"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":"1310.2416","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-09T09:53:09Z","cross_cats_sorted":[],"title_canon_sha256":"eaac09a5328ab923a7dd892f696cf14b1e2bd1781b017c72c33b762c157f334f","abstract_canon_sha256":"ba37cdbd0d0430d13d9a96d2055d7788f3184d2212b44268f99e600916f41891"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:56.838167Z","signature_b64":"oDuWl7cgK+b5pa1pZtIjl0VrjNg9dSExa3I3Sas4qbFlwgbgDRx9IK9NU2IJDL5fZ7VQlYigs/j8gUx8E9+GDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"05098f114e10690449643fd546ac55c421066ae8d6cc6441b5ef1ddbe9a90233","last_reissued_at":"2026-05-18T03:10:56.837444Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:56.837444Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Strong divisibility and lcm-sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrzej Nowicki","submitted_at":"2013-10-09T09:53:09Z","abstract_excerpt":"Let $R$ be a gcd-domain (for example let $R$ be a unique factorization domain), and let $(a_n)_{n\\geqslant1}$ be a sequence of nonzero elements in $R$. We prove that $\\gcd(a_n,a_m)=a_{\\gcd(n,m)}$ for all $n,m\\geqslant1$ if and only if $$a_n=\\prod\\limits_{d\\mid n} c_d\\quad\\mbox{for} \\ n\\geqslant1, $$ where $c_1=a_1$ and $c_n=\\mbox{lcm}(a_1,a_2,\\dots,a_n)/\\mbox{lcm}(a_1,a_2,\\dots,a_{n-1})$ for $n\\geqslant2$. 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