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A set of natural numbers that is closed under addition and whose complement in the set of non negative integers is finite is said to be a numerical semigroup. The number of positive integers not belonging to a numerical semigroup is said to be its genus. Question: How many numerical semigroups of a given genus do exist? Bras-Amorós announced that the number of numerical semigroups of genus 50 is 101090300128 and conjectured, in the same paper, that the growth of the number of numerical semigroups, by genus, behaves like the Fibonnacci sequence. Some work concerning this conjecture has been done by various authors, but it seems to be far from being solved. Other question: Among the semigroups of a given genus, which, or how many, are interesting? That is, which satisfy properties that have attracted the attention of the experts in the area? We shall see how the GAP package can be used to perform some experiments.
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