Solutions of the equation d (kn) = ϕ (ϕ (n))

Abstract

Let d(n) and ϕ (n) denote the number of positive integers dividing the positive integer n and the Euler’s phi function
representing the numbers less than and prime to n, respectively. In this paper, we determine all solutions of the equation d(n) = ϕ (ϕ (n))
and we prove that the equation d(kn) = ϕ (ϕ (n)) has a finite number of solutions for any k ≥ 1. Further, we characterize all solutions
of the last equation when k is prime.