The number of large prime divisors of consecutive integers

Abstract

Let ϵ( N) > 0 be a function of positive integers N and such that ϵ( N) → 0 and N ϵ( N) → ∞ as N → + ∞. Let Nν N ( n:…) be the number of positive integers n ≤ N for which the property stated in the dotted space holds. Finally, let g( n; N, ϵ, z) be the number of those prime divisors p of n which satisfy N Zϵ( N) ⩽ p ⩽ N ϵ ( N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, lim v N ( n : g( n; N, ϵ, z) − g( n + 1; N, ϵ z) = k) exists and we determine its actual value. The case k = 0 induced the present investigation. Our solution for this value shows that the natural density of those integers n for which n and n + 1 have the same number of prime divisors in the range (1) exists and it is positive.