$N =\sum_{k = 1}^{1000}k(\lceil\log_{\sqrt{2}}k\rceil-\lfloor\log_{\sqrt{2}}k\rfloor). $ Find $N$ for 
$$N =\sum_{k = 1}^{1000}k\left(\left\lceil\log_{\sqrt{2}}k\right\rceil-\left\lfloor\log_{\sqrt{2}}k\right\rfloor\right)\;.$$
How could you solve this problem? Are there sigma rules or anything? Thanls.

Hint:
When $k$ is a power of $2$ then $\left\lceil\log_{\sqrt{2}}k\right\rceil-\left\lfloor\log_{\sqrt{2}}k\right\rfloor = 0$ otherwise it's $1$
