KöMaL 2017 A691

Let $c\ge3$ be an integer, and define the sequence $a_1,a_2,\dots$ by the recurrence $a_1=c^2-1$, $a_{n+1}=a_n^3-3a_n^2+3$ $(n=1,2,\ldots)$. Show that for every integer $n\ge2$, the number $a_n$ has a prime divisor that does not divide any of $a_1,\ldots,a_{n-1}$.