On quotients of numerical semigroups for almost arithmetic progressions

Let $\langle A\rangle$ be the numerical semigroup generated by relatively prime positive integers $\{a_1,a_2,...,a_n\}$. The quotient of $\langle A\rangle$ with respect to a positive integer $p$ is defined by $\frac{\langle A\rangle}{p}=\{x\in \mathbb{N} \mid px\in \langle A\rangle\}$. The quotient $\frac{\langle A\rangle}{p}$ 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 Ap\'ery set of $\frac{a_1}{p}$ in $\frac{\langle A\rangle}{p}$ 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 $\langle A\rangle$ 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.