A first order sentence such that the finite Spectrum of that sentence is the prime numbers 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?

This exists due to very general results, namely that the set of primes is rudimentary. See this excellent survey on spectra: Durand et al. Fifty Years of the Spectrum Problem: Survey and New Results. The same holds true for all known "natural" number theoretic functions. Indeed, the authors remark in section 4.2 that "we are not aware of a natural number-theoretic function that is provably not rudimentary".
