There has been much work done on finding optimal packings of $k$ congruent circular discs in a larger circle; see here or here for a tabulation of many results.
Scrolling through the results, one sees that for small values of $k$, the circles are usually very tightly constrained, but for higher $k$ we often find that some of the circles in an optimal packing are untethered to any of their neighbors and can freely move around. The latter of the two links in the previous paragraph clearly lists the number of loose circles in each packing, and it appears that this configuration with $91$ circles is the largest $k$ for which the best packing has no loose circles:
Everything else seems to exhibit this behavior for higher $k$ up through $2600$ (at least, of the best-known arrangements).
What can be proven about this behavior? In particular, I am curious which of the following things have been shown about the set of positive integers $k$ such that the optimal packing of $k$ circles in a circle has loose discs:
The set is infinite.
The set has positive natural density.
The set has density $1$.
The set is cofinite.
The set contains all integers greater than $N$ for some explicitly known $N$.
I expect that all of the above conditions are true, and probably with $N=91$, but given how hard individual cases are to prove optimality for I could imagine that it is very difficult to show later items on this list. What is known in the literature about such results?
I have chosen the circle as a natural packing target for which much work appears to have been done already, but I would also be interested in analogous results for circle packing in a square (or general results about circle packing in any sufficiently non-pathological shape, for which I would expect similar kinds of statements to hold).