SMO 2013 Problem 1

Let $a_1$ , $a_2$ , ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$ , $a_{n+1}$ is the sum of the $2013$ -th powers of the digits of $a_n$ . Do there exist distinct positive integers $i$ , $j$ such that $a_i=a_j$ ?