Windows Carrying a System of Distinct Multiples

Let $f(n,m)$ be the least $$ such that the interval $(m, m+]$ contains distinct integers $a_1, , a_n$ with $i a_i$ for each $i$. The interval length is not the only variable: where the window sits matters, and the two natural specialisations behave completely differently. Minimised over the window position the answer is exactly $n$, attained just above $lcm(1,,n)$, where the $n$ consecutive integers $m+1, , m+n$ each carry the divisor they need. At the window starting at $n$ the problem is genuine, and the values are $1,2,3,5,5,8,8,10,12$ for $n 9$. The first departure from $n$ occurs at $n = 4$: the window $(4,8]$ contains only two even numbers, and three of the four indices need one.