{"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/partitions-with-prescribed-sum-of-reciprocals-1","title":"Partitions with prescribed sum of reciprocals: computational results","arxiv_id":"2502.01409","date":"2025-02-03","proceeding":null,"authors":["Wouter van Doorn"],"abstract":"For a positive rational $\\alpha$, call a set of distinct positive integers $\\{a_1, a_2, \\ldots, a_r\\}$ an $\\alpha$-partition of $n$, if the sum of the $a_i$ is equal to $n$ and the sum of the reciprocals of the $a_i$ is equal to $\\alpha$. Define $n_{\\alpha}$ to be the smallest positive integer such that for all $n \\ge n_{\\alpha}$ an $\\alpha$-partition of $n$ exists and, for a positive integer $M \\ge 2$, define $N_M$ to be the smallest positive integer such that for all $n \\ge N_M$ a $1$-partition of $n$ exists where $M$ does not divide any of the $a_i$. In this paper we determine $N_M$ for all $M \\ge 2$, and find the set of all $\\alpha$ such that $n_{\\alpha} \\le 100$.","url_abs":"https://arxiv.org/abs/2502.01409v1","url_pdf":"https://arxiv.org/pdf/2502.01409v1.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":"partitions-with-prescribed-sum-of-reciprocals-1","repo_url":"https://github.com/woett/partition-data","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}