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Title: ROUGH INTEGERS WITH A DIVISOR IN A GIVEN INTERVAL
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 $1\leq a_{2}\leq b_{2}\leq N$ .
Authors:
Award ID(s):
1802139
Publication Date:
NSF-PAR ID:
10338318
Journal Name:
Journal of the Australian Mathematical Society
Volume:
111
Issue:
1
Page Range or eLocation-ID:
17 to 36
ISSN:
1446-7887
Sponsoring Org:
National Science Foundation
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