On $\mu$-Sondow Numbers

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.