A class of solutions of the equation d(n2) = d(φ(n))

Abstract

For any positive integer $n$ let $d\left( n\right) $ and $\varphi \left( n\right) $ be the number of divisors of $n$ and the Euler's phi function of $n$, respectively. In this paper we present some notes on the equation $d\left( n^{2}\right) =d\left( \varphi \left( n\right) \right).$ In fact, we characterize a class of solutions that have at most three distinct prime factors. Moreover, we show that Dickson's conjecture implies that $d\left( n^{2}\right) =d\left( \varphi \left( n\right) \right) $ infinitely often.