About the congruence ${\sum }_{k=1}^n{k}^{f(n)}\equiv 0\text{(mod n)}$

José María Grau, Antonio M. Oller-Marcén

In this paper we characterize, in terms of the prime divisors of $n$, the pairs $(k,n)$ for which $n$ divides ${\sum }_{j=1}^n{j}^k$. As an application, we study the sets $\mathcal{M…