Let S[x] be the digital root function (also known as the repeated digital sum function), where one adds the digits of positive integer x, then adds the digits of the sum until obtaining a single-digit number. (For example, S = 3 because 9 + 7 + 5 = 21 and 2 + 1 = 3).
Given S[aa] = 2, what is the smallest positive integer that a can be
such that a is a perfect power?
Note: a is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = a.