Numerical semigroups with concentration two

Abstract: We define the concentration of a numerical semigroup $S$ as $\mathsf{C}(S)=\max \left{\text{next}_S(s)-s ~|~ s\in S \backslash {0}\right}$ wherein $\text{next}_S(s)=\min\left{x \in S ~|~ s<x\right}$. In this paper, we study the class of numerical semigroups with concentration $2$. We give algorithms to calculate the whole set of this class of semigroups with given multiplicity, genus or Frobenius number. Separately, we prove that this class of semigroups verifies the Wilf's conjecture.