The number of large prime divisors of consecutive integers

Let ?(N) > 0 be a function of positive integers N and such that ?(N) → 0 and N^?(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, ?, z) be the number of those prime divisors p of n which satisfy N^Z?(N) ? p ? N^?(N), 0 < z < 1 In the present note we show that for each k = 0, ±1, ±2,…, as N → ∞, limv_N(n : g(n; N, ?, z) ? g(n + 1; N, ?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.