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[975] = 3 because 9 + 7 + 5 = 21 and 2 + 1 = 3).

Given S[a^a] = 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 m^k = a.

(In reply to Extra Credit (spoiler) by Steve Herman)

I think extra credit would be if you can provide me the exact count of the number of digits a^a would be for each one... :-P

Posted by Dej Mar on 2008-07-10 17:47:35