Unitary harmonic numbers

Abstract

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