{"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/on-the-frobenius-number-of-certain-numerical","title":"On the Frobenius number of certain numerical semigroups","arxiv_id":"2001.11204","date":"2020-01-30","proceeding":null,"authors":["Michael Hellus","Anton Rechenauer","Rolf Waldi"],"abstract":"Let $0<\\lambda\\leq1$, $\\lambda\\notin\\left\\{\\frac24, \\frac27, \\frac2{10}, \\frac2{13}, \\ldots\\right\\}$, be a real and $p$ a prime number, with $[p,p+\\lambda p]$ containing at least two primes. Denote by $f_\\lambda(p)$ the largest integer which cannot be written as a sum of primes from $[p,p+\\lambda p]$. Then \\[f_\\lambda(p)\\sim\\left\\lfloor2+\\frac2\\lambda\\right\\rfloor\\cdot p\\text{, as }p\\text{ goes to infinity.}\\] Further a question of Wilf about the 'Money-Changing Problem' has a positive answer for all semigroups of multiplicity $p$ containing the primes from $[p,2p]$. In particular, this holds for the semigroup generated by all primes not less than $p$. The latter special case was already shown in a previous paper.","url_abs":"https://arxiv.org/abs/2001.11204v4","url_pdf":"https://arxiv.org/pdf/2001.11204v4.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":"on-the-frobenius-number-of-certain-numerical","repo_url":"https://github.com/bleiglanz/on_the_frobeniusnumber","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}