The Integer Values of ${\sum }_{i=1}^n\frac{1}{\gcd (n,i)}$

Abstract

We study the set of integers $n\le x$ for which ${\sum }_{i=1}^n\frac{1}{\gcd (n,i)}$ takes an integer value. We show that the number of such integers is $\ll x^{2/3+\epsilon }$ for all fixed $\epsilon >0$.