{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/primitive-abundant-and-weird-numbers-with","title":"Primitive abundant and weird numbers with many prime factors","arxiv_id":"1802.07178","date":"2018-02-20","proceeding":null,"authors":["Gianluca Amato","Maximilian F. Hasler","Giuseppe Melfi","Maurizio Parton"],"abstract":"We give an algorithm to enumerate all primitive abundant numbers (briefly, PANs) with a fixed $\\Omega$ (the number of prime factors counted with their multiplicity), and explicitly find all PANs up to $\\Omega=6$, count all PANs and square-free PANs up to $\\Omega=7$ and count all odd PANs and odd square-free PANs up to $\\Omega=8$. We find primitive weird numbers (briefly, PWNs) with up to 16 prime factors, improving the previous results of [Amato-Hasler-Melfi-Parton] where PWNs with up to 6 prime factors have been given. The largest PWN we find has 14712 digits: as far as we know, this is the largest example existing, the previous one being 5328 digits long [Melfi]. We find hundreds of PWNs with exactly one square odd prime factor: as far as we know, only five were known before. We find all PWNs with at least one odd prime factor with multiplicity greater than one and $\\Omega = 7$ and prove that there are none with $\\Omega < 7$. Regarding PWNs with a cubic (or higher) odd prime factor, we prove that there are none with $\\Omega\\le 7$, and we did not find any with larger $\\Omega$. Finally, we find several PWNs with 2 square odd prime factors, and one with 3 square odd prime factors. These are the first such examples.","url_abs":"http://arxiv.org/abs/1802.07178v2","url_pdf":"http://arxiv.org/pdf/1802.07178v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"primitive-abundant-and-weird-numbers-with","repo_url":"https://github.com/amato-gianluca/weirds","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}