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Has anyone seen a problem like this in the literature? There are likely more generalized versions in sieve theory, which I am willing to tackle, but I would prefer a more elementary approach if available. The problem below will ask only about multiples of integers from $[n,2n)$, but I will have to face $[n,kn)$ at some point.

Are there really about $n^2(\log 2)$ many integers in $[2n,n^2]$ which are divisible by some integer in $[n,2n)$?

At some point, a good formula will involve the floor function, but I don't want to type it out today, so will use expressions like $n^2/(n+j) - 1$ to approximate the number of multiples of $n + j$ (for integer $j$ in $[0,n)$) in the integer interval $S=[2n,n^2]$. To the first order, I can overcount to get $n(n-2)T$, where $T$ is the sum of reciprocals of integers in $[n,2n)$ ( and so get $T$ near $\log 2$ ), but now I need to worry about numbers which are multiples of both $n+i$ and $n+j$, and possibly also of $n+h$.

So the first question is how large is the second order term $n(n-2)U$, which $U$ is a sum of terms like $\gcd (n+i,n+j)/((n+i)(n+j))$? Is $U$ smaller than $1/n$?

And the second question is how much smaller are the rest of the terms which involve sums of terms like $\gcd(n+i,n+j,n+h)/((n+i)(n+j)(n+h))$? Surely there are fewer than $n$ of these multiples?

This is sort of like the Erdos multiplication table problem, but I am really interested in multiples in $[2n,n^2)$ coming from $[n,kn)$.

Gerhard "Solutions Are Preferred Over References" Paseman, 2019.12.17.

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  • $\begingroup$ This is just the multiplication table problem. There are $o(n^2)$ such numbers. $\endgroup$
    – Lucia
    Dec 17 '19 at 18:14
  • $\begingroup$ Of course U is likely bigger than 1/n, but is it like a constant times 1/n? (Maybe counting multiples of (n+i)(n+i+p)/p gives a lower bound like log log n/n to U?) Gerhard "Could Live With LogLog Error" Paseman, 2019.12.17. $\endgroup$ Dec 17 '19 at 18:19
  • $\begingroup$ @Lucia, except I am looking at entries outside the multiplication table for n by n. Or is there a reduction you see that I am not seeing? Is it a piece of a 2n by 2n table? Gerhard "Would Really Like A Reduction" Paseman, 2019.12.17. $\endgroup$ Dec 17 '19 at 18:22
  • $\begingroup$ @Lucia, Thanks to you, I just found a reference to work of Kevin Ford on this. I may update this with the reference. Gerhard "Is Getting Ready To Read" Paseman, 2019.12.17. $\endgroup$ Dec 17 '19 at 18:27
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This is a placeholder for an answer that expands on Lucia's comment. GRP 2020.04.11.

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