Suppose I have $n$ identical unit squares and I want to use them all to tile a region with minimal perimeter $p(n)$. For instance I guess $p(n^2)=4n$, by arranging them im a $n\times n$ square.
Is there an explicit formula for $p(n)$? Or sharp bounds? How do one prove equalities or bounds for this type of quantities?
I would be happy also with a recursive formula for $p(n)$, like $p(n\cdot m) = f(p(m),p(n))$, but the factorization is not unique and I don't see how to get it easily.
My intuition is that given a certain number $n$, the minimal perimeter region is a rectangle with sides of integer lengths $l$ and $m$, where $l\cdot m=n$ and $(l,m)$ is the pair of integer numbers "closest" (in some sense) to $(\sqrt{n},\sqrt{n})$, that's where number theory could play a role.
I have no clue where to start proving something along these lines, so any hint, comment or reference is welcome!