{"paper":{"title":"On the number of total prime factors of an odd perfect number","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Joshua Zelinsky","submitted_at":"2018-10-31T01:39:30Z","abstract_excerpt":"Let $N$ be an odd perfect number. Let $\\omega(N)$ be the number of distinct prime factors of $N$ and let $\\Omega(N)$ be the total number of prime factors of $N$. We prove that if $(3,N)=1$, then $ \\frac{302}{113}\\omega - \\frac{286}{113} \\leq \\Omega. $\n  If $3|N$, then $\\frac{66}{25}\\omega-5\\leq\\Omega.$ This is an improvement on similar prior results by the author which was an improvement of a result of Ochem and Rao. We also establish new lower bounds on $\\omega(N)$ in terms of the smallest prime factor of $N$ and establish new lower bounds on $N$ in terms of its smallest prime factor."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.13063","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1810.13063/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}