MathNet 2015-part-06 — 191 problems (EN–VI) Question 133

For any positive integer $n$, let $a_n = \sum_{k=1}^{\infty} \left[ \frac{n+2^{k-1}}{2^k} \right]$, where $[x]$ is the largest integer that is equal or less than $x$. Determine the value of $a_{2015}$.