DE-MO 1997 Problem 2

For a positive integer $k$ , let us denote by $u(k)$ the greatest odd divisor of $k$ . Prove that, for each $n \in N$ , $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$ .