Questions about a certain set of primes

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?

• Why do you say that $A$ "clearly" does not contain all primes? I'm pretty sure that that is a long-standing open problem. Similarly if you adjoin to $A_{n-1}$ only the smallest prime dividing $x_n$, which is maybe more in the spirit of Euclid's original proof that there are infinitely many primes. – Joe Silverman Jan 22 '19 at 3:23
• Maybe this is a bit of a tangent, but is any purpose served by writing $\left( \prod_{p_i \, \in \, A_{n-1}} p_i \right) + 1$ instead of $\left( \prod_{p \, \in \, A_{n-1}} p \right) + 1 \text{ ?} \qquad$ – Michael Hardy Jan 22 '19 at 4:10
• This sequence may be oeis.org/A126263 – Gerry Myerson Jan 22 '19 at 11:29

The related problem where only the minimum prime divisor of $$x_n$$ is added to the list $$A_{n-1}$$ is open as pointed out by @JoeSilverman. This sequence of primes is called the Euclid-Mullin sequence. See the following links:
• The Euclid-Mullin sequence is not a priori contained in the sequence in the OP, because $x_n$ is not the same. – François Brunault Jan 22 '19 at 7:01