A023414 If any power of 2 ends with k 7's and 8's, they must be the first k elements of this sequence in reverse order.

8, 8, 8, 7, 7, 8, 7, 8, 7, 8, 8, 8, 8, 8, 7, 8, 8, 8, 7, 8, 8, 8, 8, 8, 7, 7, 8, 8, 7, 8, 8, 7, 8, 7, 8, 8, 8, 7, 8, 7, 8, 7, 7, 8, 8, 7, 8, 7, 7, 7, 8, 8, 8, 8, 7, 7, 8, 7, 8, 8, 8, 7, 7, 7, 8, 7, 8, 8, 7, 7, 7, 8, 7, 7, 8, 8, 7, 8, 8, 8, 7, 8, 8, 7, 8, 7, 8, 8, 7, 7, 8, 8, 7, 8, 8, 7, 8, 8, 7, 8, 7, 8, 7, 8, 7, 8, 8, 8