Gaps between \(r\)-free integers
type: Item
instance of: scholarly article
title: Gaps between \(r\)-free integers
zbMATH Open document ID: 0805.11064
DOI: 10.1007/BF01189887
author: Catalina Calderón
author: M. J. de Velasco
published in: Archiv der Mathematik
publication date: 1994-11-13T00:00:00Z
review text: For \(r \geq 2\) let \(q_ r(n)\) be the \(n\)-th \(r\)-free integer (number not divisible by any \(r\)-th power other than 1). It is proved that \(q_ r (n+1) - q_ r(n) = O(n^{\alpha + \varepsilon})\) as \(n \to \infty\), with \(\alpha = 229/348(r+1)\). In particular, if \(q_ 2(n)\) is the \(n\)-th squarefree integer then \(\alpha = 229/1044 = 0.219348659 < 1057/4785\), that is \(q_ 2(n + 1) - q_ 2(n) = O(n^{229/1044 + \varepsilon})\).
Mathematics Subject Classification ID: 11N25
Mathematics Subject Classification ID: 11L07
zbMATH DE Number: 687055
zbMATH Keywords: exponent pair
zbMATH Keywords: distribution of integers
zbMATH Keywords: squarefree integer
MaRDI profile type: Publication
cites work: On the difference between consecutive squarefree integers
cites work: Q4692762
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cites work: Q3220660
cites work: VAN DER CORPUT'S METHOD AND THE THEORY OF EXPONENT PAIRS
cites work: On the Difference between Consecutive Squarefree Numbers
cites work: On the Gaps between Squarefree Numbers
description: scientific article; zbMATH DE number 687055