The number of addends in the decomposition of an element of a numerical semigroup into atoms

Abstract

We prove that for every nonempty set $\Sigma$ of integers bigger than $1$, which has at most three elements, there exists a numerical semigroup $T$ and an element $x$ of $T$ such that a natural number $n$ is the number of atoms in a decomposition of $x$ into atoms if and only if $n$ belongs to $\Sigma$. We also propose three related conjectures.