On a variant of Giuga numbers

TL;DR

This paper characterizes odd positive integers satisfying a specific modular sum condition and finds that their density among all positive integers is slightly over 3/8.

Contribution

It provides a new characterization of certain odd integers based on a modular sum condition and estimates their asymptotic density.

Findings

The set of integers satisfying the condition has an asymptotic density > 3/8.

Characterization of odd integers with a specific modular sum property.

The density estimate is slightly larger than 0.375.

Abstract

In this paper, we characterize the odd positive integers $n$ satisfying the congruence $\sum_{j=1} ^ {n-1} j^{\frac{n-1}{2}}\equiv 0 \pmod n$. We show that the set of such positive integers has an asymptotic density which turns out to be slightly larger than 3/8.