Abstract

A natural number n is called a k-almost prime if n has precisely k prime factors, counted with multiplicity. For any fixed k ⩾ 2, let ℱ_k(X) be the number of k-th powers m^k ⩽ X such that φ(n) = m^k for some squarefree k-almost prime n, where φ(·) is the Euler function. We show that the lower bound ℱ_k(X) ≫ X ^1/k/(log X)^2k holds, where the implied constant is absolute and the lower bound is uniform over a certain range of k relative to X. In particular, our results imply that there are infinitely many pairs (p, q) of distinct primes such that (p – 1) (q – 1) is a perfect square.