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- W4308342045 abstract "Let $S=leftlangle s_1,ldots,s_nrightrangle$ be a numerical semigroup generated by the relatively prime positive integers $s_1,ldots,s_n$. Let $kgeqslant 2$ be an integer. In this paper, we consider the following $k$-power variant of the Frobenius number of $S$ defined as $${}^{k!}r!left(Sright):= text{ the largest } k text{-power integer not belonging to } S.$$In this paper, we investigate the case $k=2$. We give an upper bound for ${}^{2!}r!left(S_Aright)$ for an infinite family of semigroups $S_A$ generated by {em arithmetic progressions}. The latter turns out to be the exact value of ${}^{2!}r!left(leftlangle s_1,s_2rightrangleright)$ under certain conditions. We present an exact formula for ${}^{2!}r!left(leftlangle s_1,s_1+d rightrangleright)$ when $d=3,4$ and $5$, study ${}^{2!}r!left(leftlangle s_1,s_1+1 rightrangleright)$ and ${}^{2!}r!left(leftlangle s_1,s_1+2 rightrangleright)$ and put forward two relevant conjectures. We finally discuss some related questions." @default.
- W4308342045 created "2022-11-11" @default.
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- W4308342045 date "2020-06-25" @default.
- W4308342045 modified "2023-09-30" @default.
- W4308342045 title "The Square Frobenius Number" @default.
- W4308342045 doi "https://doi.org/10.48550/arxiv.2006.14219" @default.
- W4308342045 hasPublicationYear "2020" @default.
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