Free Falling

Consider $\lceil \log_{\sqrt{2}}k \rceil - \lfloor \log_{\sqrt{2}}k \rfloor$ . This has a value of $0$ when $\log_{\sqrt{2}}k$ is an integer and $1$ otherwise.

When $\log_{\sqrt{2}}k$ is equal to some integer $m$ , $\log_{\sqrt{2}}k=m$ $k=2^{\frac {m}{2}}$

Since $k$ is also a positive integer, we require $m$ to be even, hence $k$ is a perfect power of 2.

Thus the desired sum is the sum of natural numbers from 1 to 1000, except for powers of 2, or $\dfrac {1000*1001}{2} - (1+2+4+\ldots+512)$ $=500500-1023=499477$

The answer is thus $\boxed{477}$ .