The number of atoms in an atomic domain

We study the number of atoms and maximal ideals in an atomic domain with finitely many atoms and no prime elements. We show in particular that for all $m,n\in {\mathbb{Z}}^{+}$ with $n\ge 3$ and $4\le m\le \frac{n}{3}$ there is an atomic domain with precisely $n$ atoms, precisely $m$ maximal ideals and no prime elements. The proofs use both commutative algebra and additive number theory.