On quotients of numerical semigroups for almost arithmetic progressions

Let $⟨A⟩$ be the numerical semigroup generated by relatively prime positive integers $\{a_1,a_2,\mathrm{.}\mathrm{.}\mathrm{.},a_n\}$. The quotient of $⟨A⟩$ with respect to a positive integer $p$ is defined by $\frac{⟨A⟩}{p}=\{x\in \mathbb{N}\mid px\in ⟨A⟩\}$. The quotient $\frac{⟨A⟩}{p}$ is known to be a semigroup but is hard to study. When $p$ is a positive divisor of $a_1$, we reduce the computation of the Apéry set of $\frac{a_1}{p}$ in $\frac{⟨A⟩}{p}$ to a simple minimization problem. This allow us to obtain closed formulas of the Frobenius number of the quotient for some special numerical semigroups. These includes the cases when $⟨A⟩$ is the almost arithmetic progressions, the almost arithmetic progressions with initial gaps, etc. In particular, we partially solve an open problem proposed by A. Adeniran et al.