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.