Prime-perfect Numbers

We discuss a relative of the perfect numbers for which it is possible to prove that there are infinitely many examples. Call a natural number n prime-perfect if n and σ(n) share the same set of distinct prime divisors. For example, all even perfect numbers are prime-perfect. We show that the count Nσ(x) of prime-perfect numbers in [1, x] satisfies estimates of the form exp((log x) log log log ) ≤ Nσ(x) ≤ x 1 3 , as x → ∞. We also discuss the analogous problem for the Euler function. Letting Nφ(x) denote the number of n ≤ x for which n and φ(n) share the same set of prime factors, we show that as x→∞, x ≤ Nφ(x) ≤ x L(x)1/4+o(1) , where L(x) = x log log x/ log log . We conclude by discussing some related problems posed by Harborth and Cohen.