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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the Schnirelmann density of the $k$-free integers


Authors: P. H. Diananda and M. V. Subbarao
Journal: Proc. Amer. Math. Soc. 62 (1977), 7-10
MSC: Primary 10L10
DOI: https://doi.org/10.1090/S0002-9939-1977-0435024-9
MathSciNet review: 0435024
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Abstract: Let ${Q_k}(n)$ be the number of k-free integers $\leqslant n$ and $d({Q_k})$ the Schnirelmann density of the k-free integers. If $k \geqslant 5$, it is shown that ${Q_k}(n)/n = d({Q_k})$ for some n satisfying ${6^k}/2 \leqslant n < {6^k}$ and certain other properties, and that \[ d({Q_k}) \geqslant 1 - {2^{ - k}} - {3^{ - k}} - {5^{ - k}} + ({3^{ - k}} + 2 \cdot {5^{ - k}}){({6^k} - {3^k} + 1)^{ - 1}}.\] $d({Q_k})$ and the n for which ${Q_k}(n)/n = d({Q_k})$ are found for $7 \leqslant k \leqslant 12$.


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Keywords: Schnirelmann density, <I>k</I>-free integers
Article copyright: © Copyright 1977 American Mathematical Society