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Rough integers with a divisor in a given interval
TLDR
In this paper, the number of distinct integers that have no prime factor and a divisor in the multiplication table is estimated in terms of the order in which distinct integers are free of prime factors.Abstract:
We determine, up to multiplicative constants, the number of integers $n\le x$ that have no prime factor $\le w$ and a divisor in $(y,2y]$. 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 $\le w$, and the number of distinct fractions of the form $\frac{a_1a_2}{b_1b_2}$ with $1\le a_1 \le b_1\le N$ and $1\le a_2\le b_2 \le N$.read more
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Journal Article
On the Greatest Prime Factor of x~n-y~n
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References
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Journal ArticleDOI
The distribution of integers with a divisor in a given interval
TL;DR: In this article, the order of magnitude of H(x;y;z), the number of integers n x having a divisor in (y,z), for all x;y and z, was established.
Posted Content
The distribution of integers with a divisor in a given interval
TL;DR: In this paper, the order of magnitude of H(x,y,z), the number of integers having a divisor in (y, z), for all x,y and z, was established.
Journal ArticleDOI
On the number of restricted prime factors of an integer. I
TL;DR: In this article, the frequencies of large deviations of co(n, E) and ~(n; E) from their normal order of magnitude were studied. But the results used in [37] were ''almost'' elementary.
Journal ArticleDOI
Some remarks on number theory. II.
TL;DR: In this article, disconnected remarks on number theory are made in the context of the paper of the same title, and the disconnected remarks are shown to be disconnected from the original paper. [5]
Journal ArticleDOI
Sur une question d'Erdös et Schinzel, II
TL;DR: In this paper, the authors define a polynomial with integer coefficients and define H_F(x,y,z) to be the number of integers not exceeding x for which F(n) has at least one divisor d such that y ≥ 0.