Numerical Semigroups with $a_e=2g+1$

Abstract

This article discusses numerical semigroups having a generator which is as large as possible. This turns out to be $2g+1$, where $g$ is the genus of the semigroup. We will show that these semigroups are closely related to symmetric semigroups and have interesting symmetry properties themselves. Furthermore we will show that Wilf's question has a positive answer for these semigroups and some semigroups derived thereof.