Distribution of genus among numerical semigroups with fixed Frobenius number

Abstract A numerical semigroup is a sub-monoid of the natural numbers under addition that has finite complement. The size its complement called genus and largest number in Frobenius number. We consider set semigroups with fixed f analyse their genus. find asymptotic distribution this show it product Gaussian power series. almost all have close to $$\frac{3f}{4}$$ 3 f 4 . denote by N ( ). While ) not monotonic we prove $$N(f)<N(f+2)$$ N ( ) < + 2 for every