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.