I'm trying to prove that any natural number $N$ can be formed by adding at least two positive consecutive integers except for powers of $2$. For example, using $\,N = 3$, $N = 1 + 2$. When experimenting with different values for $N$, I found that the there are no positive consecutive integers that when summed together give a power of $2$.
Therefore, I was wondering how you could prove that a number in the form $2^n$ cannot be formed by adding at least two positive consecutive integers.
P.S. I started by proving that all odd numbers can be formed by adding two consecutive integers: