Numerical semigroup

A numerical semigroup is a subset of the nonnegative integers that contains zero, is closed under addition, and has finite complement within the nonnegative integers. For example,Its missing values are $1,2,4,7$. To verify that no larger value is missing, observe that $8,9,10$ are present and then repeatedly add three, using consecutive-base induction for additive semigroups.

A finite family of positive integer generators produces a numerical semigroup exactly when their greatest common divisor is one. The condition is necessary because all generated integers are multiples of that greatest common divisor. Sufficiency follows by finding a generated representative of each residue class modulo one generator and then adding multiples of that generator.

Suppose an additively closed subset $M$ of the nonnegative integers contains a positive integer $a$ and every integer from $N$ to $N+a-1$. Then it contains every integer at least $N$. The first $a$ values are the base cases. For $n\ge N+a$, the smaller number $n-a\ge N$ belongs to $M$ by strong induction, and closure under addition gives $n=(n-a)+a\in M$. Equivalently, write $n=N+qa+r$ by division with remainder, with $0\le r<a$; start at $N+r\in M$ and add $a$ exactly $q$ times.