Školjka

Let $c \geq 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and $$ a_{n+1} = a_n^3 - 4c \cdot a_n^2 + 5c^2 \cdot a_n + c $$ for all $n \geq 1$. Prove that for each integer $n \geq 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1, \ldots, a_{n-1}$. \begin{flushright}\emph{(Austria)}\end{flushright}