Question 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.