{"title":"On the number of prime factors with a given multiplicity over h-free and h-full numbers","authors":"Sourabhashis Das, Wentang Kuo, Yu-Ru Liu","doi":"10.1016/j.jnt.2024.08.007","DOIUrl":null,"url":null,"abstract":"<div><div>Let <em>k</em> and <em>n</em> be natural numbers. Let <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> denote the number of distinct prime factors of <em>n</em> with multiplicity <em>k</em> as studied by Elma and the third author <span><span>[5]</span></span>. We obtain asymptotic estimates for the first and the second moments of <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> when restricted to the set of <em>h</em>-free and <em>h</em>-full numbers. We prove that <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> has normal order <span><math><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi></math></span> over <em>h</em>-free numbers, <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>h</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> has normal order <span><math><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi></math></span> over <em>h</em>-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with <span><math><mn>1</mn><mo><</mo><mi>k</mi><mo><</mo><mi>h</mi></math></span> do not have normal order over <em>h</em>-free numbers and <span><math><msub><mrow><mi>ω</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span> with <span><math><mi>k</mi><mo>></mo><mi>h</mi></math></span> do not have normal order over <em>h</em>-full numbers.</div></div>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022314X2400194X","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
Let k and n be natural numbers. Let denote the number of distinct prime factors of n with multiplicity k as studied by Elma and the third author [5]. We obtain asymptotic estimates for the first and the second moments of when restricted to the set of h-free and h-full numbers. We prove that has normal order over h-free numbers, has normal order over h-full numbers, and both of them satisfy the Erdős-Kac Theorem. Finally, we prove that the functions with do not have normal order over h-free numbers and with do not have normal order over h-full numbers.
设 k 和 n 都是自然数。让 ωk(n)表示乘数为 k 的 n 的不同质因数的个数,如 Elma 和第三作者所研究的那样[5]。我们得到了ωk(n)的第一矩和第二矩的渐近估计值,并将其限制在无 h 和满 h 的数集合中。我们证明ω1(n) 在 h 个无穷数上有正序 loglogn,ωh(n) 在 h 个满数上有正序 loglogn,而且它们都满足厄尔多斯-卡克定理。最后,我们证明含 1<k<h 的函数 ωk(n) 在无 h 数上没有正序,含 k>h 的函数 ωk(n) 在满 h 数上没有正序。