Let $A$ be a set of prime numbers, generated by the following procedure. Let $A_0 = \{2\}$. Let $A_n$ be generated by setting $x_n = (\prod_{p_i \in A_{n-1}}p_i) + 1$, and adding all the prime factors of $x_n$ to $A_{n-1}$. Let $A=\bigcup_{n\in \mathbb{N}} A_n$.

Clearly, $A$ does not contain all primes. Is there any other way to characterize the elements of $A$? What's the computational complexity of determining whether a prime belongs to $A$? What's the density of $A$ in the set of primes?