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

Abstract

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 ${\mathcal{M}}_f:=\{n:n\text{divides}{\sum }_{j=1}^n{j}^{f(n)}\}$ for some choices of $f$.

References in corpus (1)

• Generalizing Giuga's conjecture