We define the concentration of a numerical semigroup $S$ as $\mathsf{C}\left(S\right)=max\left\{{\text{next}}_S\left(s\right)-s\mid s\in S\setminus \left\{0\right\}\right\}$ wherein ${\text{next}}_S\left(s\right)=min\left\{x\in S\mid s<x\right\}$. 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.

中文翻译：

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浓度为 2 的数值半群

我们定义一个数值半群的浓度$\mathrm{小号}$作为$\mathsf{C}\left(\mathrm{小号}\right)=最大限度\left\{{\text{下一个}}_{\mathrm{小号}}\left(s\right)-s\mid s\in \mathrm{小号}\setminus \left\{0\right\}\right\}$其中${\text{下一个}}_{\mathrm{小号}}\left(s\right)=分钟\left\{X\in \mathrm{小号}\mid s<X\right\}$. 在本文中，我们研究浓度为 2 的数值半群类。我们给出算法来计算具有给定重数、属或 Frobenius 数的这类半群的整个集合。另外，我们证明了这类半群验证了 Wilf 猜想。