Which positive integers can NOT be written as a sum of consecutive positive integers

Suppose $$2^n = \sum_{i=0}^{k-1} (m+i) = km + \frac{k(k-1)}{2}$$ for some $k>1$ and $m>0$, then $2^{n+1}=2km+k(k-1)=k(2m+k-1)$. Note then that this implies that $k=2^\ell$ for some $\ell > 0$, and therefore that $2m+k-1$ is odd, and therefore that in fact, $k=2^{n+1}$, and $2m+k-1=1$. Therefore we have $2m+2^{n+1}=2$, and $m+2^n = 1$, so $m=1-2^n\le 0$. Thus $2^n$ is not expressible as a sum of consecutive positive integers.

Edit:

I realized I posted this answer without really addressing the actual question, although I sort of answered in the comments. Everything other than the argument to show that powers of $2$ can't be expressed looked ok to me. I didn't follow your argument for the powers of $2$, which is why I wrote this answer.