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ROUGH INTEGERS WITH A DIVISOR IN A GIVEN INTERVAL
Published online by Cambridge University Press: 08 January 2020
Abstract
We determine, up to multiplicative constants, the number of integers $n\leq x$ that have a divisor in $(y,2y]$ and no prime factor $\leq w$. Our estimate is uniform in $x,y,w$. We apply this to determine the order of the number of distinct integers in the $N\times N$ multiplication table, which are free of prime factors $\leq w$, and the number of distinct fractions of the form $(a_{1}a_{2})/(b_{1}b_{2})$ with $1\leq a_{1}\leq b_{1}\leq N$ and $1\leq a_{2}\leq b_{2}\leq N$.
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- Research Article
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- © 2020 Australian Mathematical Publishing Association Inc.
Footnotes
Communicated by I. Shparlinski
Research supported by National Science Foundation grant DMS-1802139.
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