On $\mu$-Sondow 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 Erdös-Moser equation and we present some conjectures about them.