<spanclass=′latex−bold′>N3:</span> For any positive integer n, define rad(n) to be the product of all prime divisors of n (without multiplicities), and in particular rad(1)=1. Consider an infinite sequence of positive integers {an}n=1∞ satisfying that
\begin{align*} a_{n+1} = a_n + rad(a_n), \: \forall n \in \mathbb{N} \end{align*}
Show that there exist positive integers t,s such that at is the product of the s smallest primes.
Proposed by ltf0501 IMOCnumber theoryprime numbers