On quotients of numerical semigroups for almost arithmetic progressions

Let [Formula: see text] be the numerical semigroup generated by relatively prime positive integers [Formula: see text]. The quotient of [Formula: see text] with respect to a positive integer p is defined by [Formula: see text]. The quotient [Formula: see text] is known to be a semigroup, but is hard to study. When p is a positive divisor of [Formula: see text], we reduce the computation of the Apéry set of [Formula: see text] in [Formula: see text] to a simple minimization problem. This allows us to obtain closed formulas of the Frobenius number of the quotient for some special numerical semigroups. These include the cases when [Formula: see text] is the almost arithmetic progressions, the almost arithmetic progressions with initial gaps. In particular, we partially solve an open problem proposed by Adeniran et al. [On the genus of a quotient of a numerical semigroup, Semigroup Forum. 98 (2019) 690–700].