On $\mu$-Sondow Numbers

TL;DR

This paper introduces and studies $mbda$-Sondow numbers, a class of integers characterized by a specific sum involving their prime divisors, relating them to Giuga numbers, weak primary pseudoperfect numbers, and the Erd
51s-Moser equation.

Contribution

It provides multiple characterizations of $mbda$-Sondow numbers, connects them to existing number classes, and explores conjectures and relations to the Erd
51s-Moser equation.

Findings

Characterizations of $mbda$-Sondow numbers similar to Giuga numbers

Relation established between $mbda$-Sondow numbers and Erd
51s-Moser equation

Introduction of conjectures about the properties of these numbers

Abstract

Given an integer $\mu$, we study the numbers that satisfy the condition $\frac{\mu}{n} + \sum_ {p \mid n} \frac {1} {p} \in \mathbb{N}$. This condition, which is reminiscent of the one satisfied by Giuga numbers ($\mu=-1$), also includes the so-called \cite{sondow} weak primary pseudoperfect numbers ($\mu=1$). As a tribute to our late colleague Jonathan Sondow (1943 -- 2020), we have named these numbers $ \mu $-Sondow numbers. In this paper, we give several different characterizations of these numbers, all of them suggested by well-known characterizations of the Giuga numbers. We also relate these numbers to the well-known Erd\"os-Moser equation and we present some conjectures about them.