$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: If $k=2^n$, then $\log_{\sqrt2}k=2n$, and therefore
$$\left\lceil\log_{\sqrt2}k\right\rceil=\left\lfloor\log_{\sqrt2}k\right\rfloor\;.\tag{1}$$


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*Are there any other values of $k$ for which $(1)$ is true?  

*When $(1)$ is false, what is $\left\lceil\log_{\sqrt2}k\right\rceil-\left\lfloor\log_{\sqrt2}k\right\rfloor$

