By $rad(x)$ we denote the product of all distinct prime factors of a positive integer $n$. Given $a \in N$, a sequence $(a_n)$ is defined by $a_0 = a$ and $a_{n+1} = a_n+rad(a_n)$ for all $n \ge 0$. Prove that there exists an index $n$ for which $\frac{a_n}{rad(a_n)} = 2022$