Question 59
For any positive integer $n$ let $a_n$ be the largest power of $2$ that divides $n$ (e.g. $a_{2011} = 1$, $a_{2012} = 4$). Prove that for any positive integers $i$ and $j$ with $i \lt j$, the sum $\frac{1}{a_i} + \frac{1}{a_{i+1}} + \dots + \frac{1}{a_j}$ is a fractional number.