Inspired by the delicious buns and Siu Mai in bamboo steamers I saw tonight in a food show about Cantonese Dim Sum, here is a natural question. It probably has been well studied in the literature, but I cannot find related reference.

Given a real number $r \le 1$, let $f(r)$ be the maximum number of radius-$r$ disks that can be packed into a unit disk. For example, $f(1)=1$ for $r \in (1/2, 1]$, $f(r)=2$ for $r \in (2\sqrt{3}-3, 1/2]$, etc.

**Question:** Is it true that $\{f(r): r \in (0, 1]\}=\mathbb{N}$?

Noam Elkies immediately pointed out that the maximum radius allowed for packing $6$ and $7$ balls are the same (both $1/3$). Now let me change the problem to just make it harder :)

**Modified question:** Is $\mathbb{N} \setminus \{f(r): r \in (0, 1]\}$ a finite set?