CN-SMO 2015 Problem 4

For any positive integer $n$ , we have the set $P_n = \{ n^k \mid k=0,1,2, \ldots \}$ . For positive integers $a,b,c$ , we define the group of $(a,b,c)$ as lucky if there is a positive integer $m$ such that $a-1$ , $ab-12$ , $abc-2015$ (the three numbers need not be different from each other) belong to the set $P_m$ . Find the number of lucky groups.