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Proceedings of the American Mathematical Society

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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

$\displaystyle d({Q_k}) \geqslant 1 - {2^{ - k}} - {3^{ - k}} - {5^{ - k}} + ({3^{ - k}} + 2 \cdot {5^{ - k}}){({6^k} - {3^k} + 1)^{ - 1}}.$

$ 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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DOI: https://doi.org/10.1090/S0002-9939-1977-0435024-9
Keywords: Schnirelmann density, k-free integers
Article copyright: © Copyright 1977 American Mathematical Society