{"title":"Powerfree sums of proper divisors","authors":"P. Pollack, A. Roy","doi":"10.4064/cm8616-10-2021","DOIUrl":null,"url":null,"abstract":". Let s ( n ) := (cid:80) d | n,d<n d denote the sum of the proper divisors of n . It is natural to conjecture that for each integer k ≥ 2 , the equivalence n is k th powerfree ⇐⇒ s ( n ) is k th powerfree holds almost always (meaning, on a set of asymptotic density 1 ). We prove this for k ≥ 4 .","PeriodicalId":49216,"journal":{"name":"Colloquium Mathematicum","volume":" ","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2021-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Colloquium Mathematicum","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.4064/cm8616-10-2021","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS","Score":null,"Total":0}
． 让s (n) = (cid): 80) d | n, d < n d合适divisors》denote the sum of n。自然是每到conjecture that for《equivalence n是整数k≥2,k th powerfree⇐⇒s (n)是k th powerfree珍藏几乎总是(asymptotic密度之意义,on a组1)。我们证明这个for k≥4。
期刊介绍：
Colloquium Mathematicum is a journal devoted to the publication of original papers of moderate length addressed to a broad mathematical audience. It publishes results of original research, interesting new proofs of important theorems and research-expository papers in all fields of pure mathematics.
Two issues constitute a volume, and at least four volumes are published each year.
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