The Frobenius problem for a class of quotients of numerical semigroups

Abstract

Given a numerical semigroup $S$ and a positive integer $p$, the quotient $\frac{S}{p}=\{x\in \mathbb{N}\mid px\in S\}$ also forms a numerical semigroup. In this paper, we first characterize the Apéry set for a class of quotients of numerical semigroups. Under certain conditions, we then derive half-closed form formulas for their Frobenius number and genus. Furthermore, for specific values of part parameters, we obtain explicit formulas for the Frobenius number of certain quotients of numerical semigroups.

12 pages