Abstract
For a function f : N -> N, define N-f(x) (x) = #{n < x : n = kf(k) for some k}. Let tau (n) = sigma(d|n) 1 be the divisor function, omega (n) = sigma p|n 1 be the prime divisor function, and ?(n) = #{1 <= k <= n : (k, n) = 1} be Euler's totient function. We prove that (1) Nxx tau (x) <^> (log x)1/2 ; (2) N omega x (x) = (1+ o(1)) x log log x; (3) N?x (x) = (c0 + o(1))x1/2,where c0 = 1.365.. ..