Question 163
Let $a_{1}=1$, $a_{2}=2$, and for $n \geq 3$, let $a_{n}$ be the smallest positive integer such that $a_{n} \neq a_{i}$ for $i\lt n$ and $\operatorname{gcd}\left(a_{n}, a_{n-1}\right)\gt 1$. Prove that every positive integer appears as some $a_{i}$.