{"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 $n\\geqslant2$. All equalities with gcd and lcm are determined up to units of $R$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2416","kind":"arxiv","version":1},"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"}