Hello so I am working on a proof to try and describe the set of numbers which can be written as the sum of $3$ or more positive consecutive integers, I have come up with a formula for a number $$S = kn + \frac{k(k-1)}{2}$$ where $n\geq 1$ and $k\geq 3$ which tells me how to find whether a number can be written this way.
Though, I am not sure where I need to go from here to show which types of numbers will satisfy the equation..
We have $$\sum_{j=a}^{a+n} j=\frac{(n+1)(2a+n)}{2}$$
Hence a number $N$ satisfies the given property if and only if there are integers $a\ge 1$ and $n\ge 2$ , such that $2N=(n+1)(2a+n)$
So, $2N$ must be the product of two integers greater than $2$, from which one is odd and one is even. For every such number, we can find suitable $n$ and $a$.
If $N$ is a power of $2$ or a prime, this is obviously impossible. If $N$ is composite and not a power of $2$, there is an odd prime dividing $N$, denote it with $p$, and $\frac{N}{p}$ is at least $2$, so we always get the desired product.
Therefore the desired property is satisfied exactly for all compositie numbers which are not a power of $2$.