So I am working on a proof to try and describe the numbers that can be written as a sum of 3 or more positive consecutive integers so far I have come up with a formula S = kn + k(k-1)/2 where n>=1 and k>=3
But I am not sure where to go from here I need to now show which types of numbers will be satisfied by the equation but I'm not quite sure how.
$k+(k+1)+\dots+(k+n-1)=\frac{n(2k+n-1)}{2}$.
If $n$ is odd you get all of the multiples of $n$ and if $n$ is even you get all of the numbers of the form $(n/2)\times k$ where $k$ is odd.
If $n$ has an odd prime factor $p$ then it is a sum of $p$ consecutive integers.
If is an even power of $2:n=2^a$ or $n=-2^a$ then $n$ is the sum of $2^{a+1}$ consecutive integers.
So the only ones that are not of this form are $1$ and $-1$.
Any integer $s \ge 2$ will do, with $k = 2 s$ and $n = 1-s$. That is,
$$ (1-s) + (2-s) + \ldots + s = s$$
Or do you want a sum of consecutive positive integers?