A note on variants of Euler's $\varphi$-function

It is well-known that the sum of the first $n$ consecutive integers always divides the $k$-th power sum of the first $n$ consecutive integers when $k$ is odd. Motivated by this result, in this note we study the divisibility properties of the power sum of positive integers that are coprime to $n$ and not surpassing $n$. First, we prove a finiteness result for our divisibility sets using smooth numbers in short intervals. Then, we find the exact structure of a certain divisibility set that contains the orders of these power sums and our result is of computational flavour.