The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$

[2304.09039] The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$ Skip to main content Search Submit Donate Log in Search arXiv Press Enter to search · Advanced search Mathematics > Combinatorics arXiv:2304.09039 (math) [Submitted on 18 Apr 2023 ( v1 ), last revised 26 Apr 2023 (this version, v2)] Title: The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$ Authors: Feihu Liu , Guoce Xin , Suting Ye , Jingjing Yin View a PDF of the paper titled The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$, by Feihu Liu and 3 other authors View PDF HTML (experimental) Abstract: Given relative prime positive integers $A=(a_1, a_2, ..., a_n)$, the Frobenius number $g(A)$ is the largest integer not representable as a linear combination of the $a_i$'s with nonnegative integer coefficients. We find the ``Stable" property introduced for the square sequence $A=(a,a+1,a+2^2,\dots, a+k^2)$ naturally extends for $A(a)=(a,ha+dB)=(a,ha+d,ha+b_2d,...,ha+b_kd)$. This gives a parallel characterization of $g(A(a))$ as a ``congruence class function" modulo $b_k$ when $a$ is large enough. For orderly sequence $B=(1,b_2,\dots,b_k)$, we find good bound for $a$. In particular we calculate $g(a,ha+dB)$ for $B=(1,2,b,b+1)$, $B=(1,2,b,b+1,2b)$, $B=(1,b,2b-1)$ and $B=(1,2,...,k,K)$. Our idea also applies to the case $B=(b_1,b_2,...,b_k)$, $b_1> 1$. Subjects: Combinatorics (math.CO) ; Number Theory (math.NT) Cite as: arXiv:2304.09039 [math.CO] (or arXiv:2304.09039v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2304.09039 Focus to learn more arXiv-issued DOI via DataCite Related DOI : https://doi.org/10.1007/s11139-024-00837-2 Focus to learn more DOI(s) linking to related resources Submission history From: FeiHu Liu [ view email ] [v1] Tue, 18 Apr 2023 14:57:58 UTC (13 KB) [v2] Wed, 26 Apr 2023 14:29:07 UTC (16 KB) Full-text links: Access Paper: View a PDF of the paper titled The Frobenius Formula for $A=(a,ha+d,ha+b_2d,...,ha+b_kd)$, by Feihu Liu and 3 other authors View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2023-04 Change to browse by: math math.NT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Loading... BibTeX formatted citation × loading... Data provided by: Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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