The most common digit in the decimal form of 2^n; the least one if there are duplicates.
1, 2, 4, 8, 1, 2, 4, 1, 2, 1, 0, 0, 0, 1, 1, 2, 5, 1, 2, 2, 0, 2, 4, 8, 7, 3, 6, 1, 4, 0, 1, 4, 9, 9, 1, 3, 6, 3, 4, 8, 1, 2, 1, 0, 1, 8, 7, 3, 6, 2, 2, 2, 9, 9, 8, 6, 7, 1, 1, 3, 6, 3, 1, 2, 4, 3, 6, 2, 5, 5, 1, 2, 6, 9, 8, 7, 3, 1, 3, 0, 1, 2, 8, 6, 1, 6, 1
0
Sequence S000166 gives the number of such digits in 2^n.
T. D. Noe, Plot of 1001 terms
T. D. Noe, Table of 1001 terms
(Mma) Table[s = RotateRight[DigitCount[2^n]]; Position[s, Max[s], 1, 1][[1, 1]] - 1, {n, 0, 100}]
nonn,base
T. D. Noe, Jul 25 2014