Finding the Frobenius number and the genus of any numerical semigroup $S$ is a well-known open problem. Similarly, it has been studied how to express the Frobenius number and the genus of a quotient of a numerical semigroup. In this paper, by enumerating the Hilbert series of each type of numerical semigroup, we show an expression for the genus of a quotient of numerical semigroups generated by one of the following series: arithmetic progression, geometric series, and Pythagorean triple.

中文翻译：

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几种数值半群的商的属

查找任何数值半群的弗罗贝尼乌斯数和亏格$S$是一个众所周知的开放性问题。类似地，已经研究了如何表达数值半群的弗罗贝尼乌斯数和商的属。在本文中，通过枚举每种类型的数值半群的希尔伯特级数，我们展示了由以下级数之一生成的数值半群的商的表达式：算术级数、几何级数和毕达哥拉斯三元组。