On the number of prime factors with a given multiplicity over h-free and h-full numbers

2024/09/17 by Sourabhashis Das, Wentang Kuo, Das, Sourabhashis +3 · 2 citations
Mathematics · #11N05 #11N37 #Advanced Mathematical Theories #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2409.11275

openalex publication_date 2024/09/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let k and n be natural numbers. Let ω_k(n) denote the number of distinct prime factors of n with multiplicity k as studied by Elma and the third author. We obtain asymptotic estimates for the first and the second moments of ω_k(n) when restricted to the set of h-free and h-full numbers. We prove that ω₁(n) has normal order log log n over h-free numbers, ω_h(n) has normal order log log n over h-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions ω_k(n) with 1 < k < h do not have normal order over h-free numbers and ω_k(n) with k > h do not have normal order over h-full numbers.

Cited by

• On the distribution of the total number of generators of h-free and h-full elements in an abelian monoid
• On the distribution of the number of distinct generators of h-free and h-full elements in an abelian monoid

Related

• Distribution of ω(n) over h-free and h-full numbers
• Generalizations of Erdős-Kac theorem with applications
• Generalizations of the Erdős-Kac Theorem and the Prime Number Theorem
• On inequalities involving counts of the prime factors of an odd perfect number
• A note on the normal largest gap between prime factors