On the exponents in the factorizations of r consecutive numbers

Let f (n) be the number of distinct exponents in the prime factorization of the natural number n. For every r-tuple of positive integers k = (k 1,.., k r) and for all x > 1, let (Figure presented.) be the set of natural numbers n ≤ x such that f (n+i−1) = ki for i = 1,.., r. We prove that (Figure presented.) where A k ≥ 0 depends only on k and αr ∈ (0, 1) depends only on r. Moreover, we provide a characterization of the k’s such that A k > 0. This extends a previous result of the author, who considered the case r = 1.