On the Frobenius number of certain numerical semigroups
Michael Hellus, Anton Rechenauer, Rolf Waldi
备注：
15 pages, 2 figures
发表信息：
Electronic version of an article published as [International Journal of Algebra and Computation, Vol. 31, No. 03, pp. 519-532 (2021)] [copyright World Scientific Publishing Company] [https://www.worldscientific.com/worldscinet/ijac]
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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