Thinking about this more... and just thinking out loud here. So this pattern essentially happens when: In whatever base you're in a number x^n gives an end of "0" plus a remainder of the number x.
So a number would be automorphic if ((x^n - 1) * n) always ends in "0" (to whatever length that matches the number).
E.g. ((6^n - 1) * 6) or ((376^n - 1) * 376)
Cool