The Frobenius problem for a class of quotients of numerical semigroups

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.