On arithmetic numbers

TL;DR

This paper characterizes arithmetic numbers, integers whose divisor mean is integral, using cyclotomic polynomial properties, and also provides a new characterization of Mersenne numbers.

Contribution

It introduces a novel characterization of arithmetic numbers based on cyclotomic polynomial factorization and offers a new perspective on Mersenne numbers.

Findings

Arithmetic numbers are characterized via cyclotomic polynomial properties.

A new characterization of Mersenne numbers is provided.

The approach links divisor means to polynomial factorization.

Abstract

An integer $n$ is said to be \textit{arithmetic} if the arithmetic mean of its divisors is an integer. In this paper, using properties of the factorization of values of cyclotomic polynomials, we characterize arithmetic numbers. As an application, in Section 2, we give an interesting characterization of Mersenne numbers.