TITLE: Packing circles with radii 1, 2, 3, ..., n in a rectangle
QUESTION [5 upvotes]: For each positive integer n, let $a_n$ be the area of the smallest rectangle whose area is a whole number, and inside which it is possible to pack all n circles of radii 1, 2, 3, ..., n respectively (with no overlaps). 
Is it possible to determine $a_n$ precisely?
For example $a_{12}$ is at most 2466 (https://puzzling.stackexchange.com/questions/92949/my-mothers-dish-collection), and can perhaps be proved to be precisely that.

REPLY [14 votes]: Here's a better solution for $n=12$, with area approximately 2496:

Even better, with area approximately 2463:

Here's @MattF's suggestion, but it's worse in both dimensions:

@GerhardPaseman, if I consider only circles 6 through 12, this is the best solution I have found: