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 $1\leq a_{2}\leq b_{2}\leq N$ .

中文翻译：

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给定区间中带除数的粗整数

我们确定，直到乘法常数，整数的数量 $n\leq x$ 有一个除数的 $(y,2y]$ 并且没有主要因素 $\leq w$ . 我们的估计是一致的 $x,y,w$ . 我们应用它来确定不同整数的数量的顺序 $N\次N$ 没有素因数的乘法表 $\leq w$ ，以及形式的不同分数的数量 $(a_{1}a_{2})/(b_{1}b_{2})$ 和 $1\leq a_{1}\leq b_{1}\leq N$ 和 $1\leq a_{2}\leq b_{2}\leq N$ .