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arXiv · 2001.11204

On the Frobenius number of certain numerical semigroups

Michael Hellus·Anton Rechenauer·Rolf Waldi
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Abstract

Let $0<λ\leq1$, $λ\notin\left\{\frac24, \frac27, \frac2{10}, \frac2{13}, \ldots\right\}$, be a real and $p$ a prime number, with $[p,p+λp]$ containing at least two primes. Denote by $f_λ(p)$ the largest integer which cannot be written as a sum of primes from $[p,p+λp]$. Then \[f_λ(p)\sim\left\lfloor2+\frac2λ\right\rfloor\cdot p\text{, as }p\text{ goes to infinity.}\] 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.

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Michael Hellus, Anton Rechenauer, Rolf Waldi. 2022-03-01. On the Frobenius number of certain numerical semigroups. https://doi.org/10.1142/s0218196721500259

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