On the Distance between Smooth Numbers

Let P (n) stand for the largest prime factor of n ≥ 2 and set P (1) = 1. For each integer n ≥ 2, let δ(n) be the distance to the nearest P (n)-smooth number, that is, to the nearest integer whose largest prime factor is no larger than that of n. We provide a heuristic argument showing that � n≤x 1/δ(n) = (4 log 2 − 2 + o(1))x as x → ∞. Moreover, given an arbitrary real-valued arithmetic function f , we study the behavior of the more general function δf (n) defined by δf (n) = min1≤m�=n, f(m)≤f(n) |n − m| for n ≥ 2, and δf (1) = 1. In particular, given any positive integers a < b, we show that � a≤n<b 1/δf (n) ≥ 2(b − a)/3 and that if f(n) ≥ f(a) for all n ∈ [a, b[, then � a<n<b δf (n) ≤ (b− a) log(b− a)/(2 log 2).