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 = (k₁, . . . , k_r) and for all x > 1, let be the set of natural numbers n ≤ x such that f (n+i−1) = k_i for i = 1, . . . , r. We prove that

where A_k ≥ 0 depends only on k and α_r ∈ (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.

中文翻译：

---

关于 r 个连续数的因式分解中的指数

摘要

令f ( n ) 为自然数n的素数分解中不同指数的数量。对于每个正整数k = ( k ₁ , . . . , k r ) 的_r元组和所有x > 1，令是自然数集n ≤ x使得f ( n + i− 1) = k _i对于i = 1, . . . , r . 我们证明

其中A _k ≥ 0 仅取决于k并且α _r ∈ (0, 1) 仅取决于r。此外，我们提供了k的表征，使得A _k > 0。这扩展了作者先前的结果，他考虑了r = 1 的情况。