Numerical semigroups with fixed multiplicity and concentration

Abstract

We define the concentration of a numerical semigroup S as C(S)=max{nextS(s)−s∣s∈S\{0}}, where nextS(s)=min{x∈S∣s<x}. We study the class of numerical semigroups with multiplicity m and concentration at most k, denoted by Ck[m]. We give algorithms to calculate the whole set Ck[m] with given genus or Frobenius number. In addition, we prove that if S∈Ck[m] with k≤√m2, then S satisfies Wilf’s conjecture.