Let's say those $n$ integer points are $a_1,a_2,\ldots a_n$ It is clear that we must have $$a_i-0|n$$So, $n$ must have at least $\frac{n}{2}$ positive factors. It looks like the only possible $n$ are $n=1,2,3,4,6,8$. I'm not sure how to rigorously prove the last part, but the number of factors relatively compared to $n$ decreases quite rapidly and is probably maximized in terms of efficiency when $n$ is a power of $2$. @below that makes sense.

I agree mostly with @above, but not the statement that it is maximized when $n$ is a power of $2$. It should be maximized when its the product of a large number of primes.

sixoneeight wrote: I agree mostly with @above, but not the statement that it is maximized when $n$ is a power of $2$. It should be maximized when its the product of a large number of primes. yes i agree @above