Numbers of the form k + f(k)

For a function f:N→N, letNf+(x)={n⩽x:n=k+f(k) for some k}. Let τ(n)=∑d|n1 be the divisor function, ω(n)=∑p|n1 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≪Nω+(x),(2)x≪Nτ+(x)⩽0.94x,(3)x≪Nφ+(x)⩽0.93x.