The Set of Numerical Semigroups with Frobenius Number Belonging to a Fixed Interval

Abstract

Let a and b be positive integers such that (Formula presented.) and (Formula presented.) In this work, we will show that (Formula presented.) is a numerical semigroup whose Frobenius number belongs to (Formula presented.) and is a covariety. This fact allows us to present an algorithm which computes all the elements from (Formula presented.) We will prove that (Formula presented.) has multiplicity (Formula presented.) and is a ratio-covariety. As a consequence, we will show an algorithm which calculates all the elements belonging to (Formula presented.) Based on the above results, we will develop an interesting algorithm that calculates all numerical semigroups with a given multiplicity and complexity.