Frobenius Number
The largest nonnegative integer not belonging to a numerical semigroup S is the Frobenius number of S. If S is the set of nonnegative integers, then clearly its Frobenius number is -1, otherwise its Frobenius number coincides with the maximum of the gaps (or fundamental gaps) of S. An integer z is a pseudo-Frobenius number of S if z+S\setminus\{0\}\subseteq S.

NS
is a numerical semigroup. It returns the Frobenius number of NS. Of course, the time consumed to return a result may depend on the way the semigroup is given or on the knowledge already produced on the semigroup. gap> FrobeniusNumberOfNumericalSemigroup(NumericalSemigroup(3,5,7)); 4
This is just a synonym of .
S
is a numerical semigroup. It returns set of pseudo-Frobenius numbers of S. gap> S := NumericalSemigroup("modular", 5,53); <Modular numerical semigroup satisfying 5x mod 53 <= x > gap> PseudoFrobeniusOfNumericalSemigroup(S); [ 21, 40, 41, 42 ]