2006 AIME I Problem 13

For each even positive integer x, let g(x) denote the greatest power of 2 that divides x. For example, g(20)=4 and g(16)=16. For each positive integer n, let S_n=\sum_{k=1}^{2^{n-1}}g(2k). Find the greatest integer n less than 1000 such that S_n is a perfect square.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.