This question is inspired by Bernardo Recaman's question Strings of consecutive integers divisible by 1, 2, 3, ..., N on intervals of $n$ integers being divisible by the integers $1$ through $n$. The first part is to ask (i.e. help me remember) which paper(s) of Erdos considered questions like Bernardo's and the ones below.
I am (on my way to) counting (really, determining the density) of those integers $m$ such that there is a bijection $d()$ from the integers in $[m+1,m+n]$ to $[1,n]$ which satisfy $d(l)$ divides $l$ for all $l$ in the domain. (This touches on but is different from the concept of a Grimm map; search for my post on Grimm machines and related for more of this idea.) As a start on this project, I try to reduce the work by ignoring $m$ which would have two or more numbers coprime to $P(n)$ (the product of the primes at most $n$) in the domain interval, as this would preclude such a $d()$. For want of a better name (or insufficient caffeine), I call this condition local coprimality: Given $n$, $m$ is locally coprime (mod $P(n)$) if two or more numbers from $[m+1,m+n]$ are coprime to $P(n)$.
Given $m$, there is another obvious obstruction to the existence of such a map $d()$: there is a number $K$ with $m+1 \lt K \lt m+n$ such that both $n-1$ and $n$ divide $K$. Such a $K$ is the only chance for a pre-image of both $n$ and $n-1$ under a bijection of this kind, and so there can't be such a $K$ in the 'middle' of the domain interval. (If $K$ is at the end, the other end may have a multiple of $n-1$ and we may get a bijection after all.)
We can generalize this obstruction from $j=2$ up to $j \lt n$ as follows. Say $m$ is $j$-obstructing if there is a subset $S$ of $j$ many integers from $[1..n]$ such that the subset of $[m+1,m+n]$ of numbers which are divisible by any member of $S$ has strictly fewer than $j$ members in it. For example (from my post to Bernardo's question), for $n=5$ any $m$ which is 17 mod 60 is $2$-obstructing because the associated interval of 5 numbers has a multiple of 20 in the middle.
Now given $n$, there may be more than one $j \gt 1$ for which a particular $m$ is $j$-obstructing. Pick the least such $j$ (and call it $k$) and say $m$ is $k$-blocking. It is clear that if $m$ is locally coprime then $m$ is $j$-obstructing for some $j \lt n$, and therefore $m$ is $k$-blocking for some $k$.
Question: what function $k(n)$ exists that is smallest and that guarantees ($m$ is $k(n)$-blocking implies $m$ is locally coprime mod $P(n)$)? (Assume $n$ larger than 3 for interest.)
If I knew the answer to this (and a few other questions), I could write a more efficient program to determine (or at least estimate) the density of $m$ for which the desired bijection $d()$ exists.
Gerhard "Trying To Conquer The Divide" Paseman, 2019.04.16.
