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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 $A=\cup_{n\in \mathbb{N}} A_n$.

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?

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?

Let $A$ be a set of prime numbers, generated by the following procedure. Let $A_0 = \{2\}$. Let $A_n$ be generated by setting $x$ equal to the product of all primes in $A_{n-1}$, plus $1$, and adding all the prime factors of $x$ to $A_{n-1}$ (that is, factors of $x$ which were not previously members of $A_{n-1}$). Let $A=\cup_{n\in \mathbb{N}} A_n$.

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?

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 $A=\cup_{n\in \mathbb{N}} A_n$.

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?

Let A be a set of prime numbers, generated through the following procedure. Let A₀ = {2}. Let A_n be generated by setting x equal to the product of all primes in A_n-1, plus 1, and adding all the prime factors of x to A_n-1 (that is, factors of x which were not previously members of A_n-1). Let A be the union of all the 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?

Let $A$ be a set of prime numbers, generated by the following procedure. Let $A_0 = \{2\}$. Let $A_n$ be generated by setting $x$ equal to the product of all primes in $A_{n-1}$, plus $1$, and adding all the prime factors of $x$ to $A_{n-1}$ (that is, factors of $x$ which were not previously members of $A_{n-1}$). Let $A=\cup_{n\in \mathbb{N}} A_n$.

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?