The automaticity of the set of primes

The automaticity $A(x)$ of a set $\mathcal{X}$ is the size of the smallest automaton that recognizes $\mathcal{X}$ on all words of length $\leq x$. We show that the automaticity of the set of primes is at least $x\exp\left(-c(\log\log x)^2\log\log\log x\right)$, which is fairly close to the maximal automaticity.