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 $a_i=a_j$?
Source: SMO(O) 2013 #1
Tags: number theory unsolved, number theory, Sequence
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 $a_i=a_j$?