On the exponents in the factorizations of r consecutive numbers

Abstract

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 be the set of natural numbers n ≤ x such that f (n+i−1) = ki for i = 1, . , r. We prove that 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.