Search

For a function f : N → N , let N f + (x) = { n ⩽ x : n = k + f (k) for some k }. Let τ (n) = ∑ d | n 1 be the divisor function, ω (n) = ∑ p | n 1 be the prime divisor function, and φ (n) = # { 1 ⩽ k ⩽ n : gcd ⁡ (k , n) = 1 } be Euler's totient function. We show that (1) x ≪ N ω + (x) , (2) x ≪ N τ + (x) ⩽ 0.94 x , (3) x ≪ N φ + (x) ⩽ 0.93 x. [ABSTRACT FROM AUTHOR]