The finite spectrum of a theory $T$ is the set of natural numbers such that there exists a model of that size. That is $Fs(T):= \{n \in \mathbb{N} | \exists \mathcal{M}\models T : |\mathcal{M}| =n\}$ . What I am asking for is a finitely axiomatized $T$ such that $Fs(T)$ is the set of prime numbers.
In other words in what specific language $L$, and what specific $L$-sentence $\phi$ has the property that $Fs(\{\phi\})$ is the set of prime numbers?