Unitary harmonic numbers

Abstract:

If ${d^ \ast }(n)$ and ${\sigma ^ \ast }(n)$ denote the number and sum, respectively, of the unitary divisors of the natural number $n$ then the harmonic mean of the unitary divisors of $n$ is given by ${H^ \ast }(n) = n{d^ \ast }(n)/{\sigma ^ \ast }(n)$. Here we investigate the properties of ${H^ \ast }(n)$, and, in particular, study those numbers $n$ for which ${H^ \ast }(n)$ is an integer.