About the congruence $\sum_{k=1}^n k^{f(n)} \equiv 0 \textrm{(mod n)} $

TL;DR

This paper characterizes when the sum of powers of integers up to n is divisible by n, based on prime divisors, and explores related sets defined by such divisibility conditions.

Contribution

It provides a prime divisor-based characterization of divisibility conditions for sums of powers and studies the structure of sets defined by these conditions.

Findings

Characterization of divisibility in terms of prime divisors

Description of sets $\\mathcal{M}_f$ for specific functions $f$

Insights into the structure of power sum divisibility sets

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