# Decomposing a subset of $\mathbf Z$ into a sumset of irreducibles

We say that a subset $A$ of $\mathbf Z$ is irreducible if $|A| \ge 2$ and there do not exist $X, Y \subseteq \mathbf Z$ with $|X|, |Y| \ge 2$ such that $A = X + Y$.

If $X \subseteq \mathbf Z$, we denote by $\mathsf L(X)$ the set of all $k \in \mathbf N^+$ such that $X = A_1 + \cdots +A_k$ for some irreducible sets $A_1, \ldots, A_k \subseteq \mathbf Z$.

Q. Do there exist a finite set $X \subseteq \mathbf Z$ such that $|\mathsf L(X)| \ge 2$ and $2 \notin \mathsf L(X)$?

Without loss of generality, one can always restrict to considering decompositions of subsets of $\mathbf N$ containing $0$ into irreducible subsets of $\mathbf N$. So, a brute-force search shows that the answer to the question is no for all sets $X$ contained in an interval of width $\le 17$ (I don't have enough computing power to test any larger interval).

Besides that, it is perhaps worth mentioning that, for every $n \ge 2$, $\mathsf L([0, n]) = [2, n]$ and there exist finite sets $X, Y \subseteq \mathbf Z$ with $\mathsf L(X) = \{n\}$ and $\mathsf L(Y) = \{2, n\}$.