We all know formula $n(n+1)/2$ for adding up the numbers from $1$ to $n$.
But I would like to know if there is any significance and use of formulas of type $n(n^{p-1}+p-1)/p$, where $p$ is a prime. Notice that the above mentioned summation formula is just a case where $p=2$.
It is easy to show that the numerator of the general formula is indeed divisible by $p$, as $$ \frac{n(n^{p-1}+p-1)}{p} = \frac{n^p+pn-n}{p}. $$ From this we see that $pn$ is divisible by $p$, and Fermat's Little Theorem says that $(n^p-n)$ is also divisible by $p$.