Surjectivity of a recursive sequence over $\mathbb{N}$.

Question

Let $a_{0} = 2$, for $ n \geq 1$ define $a_{n} = \text{min}\{ m \in \mathbb{N} : \text{gcd}(a_{n-1} , m) \neq 1 \hspace{0.3cm} \wedge \hspace{0.3cm} \forall k \leq n-1 \hspace{0.2cm} ( m \neq a_{k})\} $.

Is it true that each $ m \geq 2$ occurs in this sequence?

Answer 1

This is known as the EKG sequence (they define $a_1 = 1, a_2 = 2$ as starting point instead) and Lagarias, Rains and Sloane prove it is a permutation of the integers.