ISL 2014 N7

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