It is [not computable] in the sense that [there is no program which, given an input n, outputs the first n digits, and then halts]. However, the set of rationa

It is [not computable] in the sense that [there is no program which, given an input n, outputs the first n digits, and then halts]. However, the set of rational numbers less than it, is computationally enumerable, in the sense that there is a program which produces an endless non decreasing sequence of rational numbers which will eventually exceed any rational number less than it. You say “using N states”. This could work, but I think the standard approach is to instead just say to run all programs in a dovetail fashion, and repeatedly count, say, which of the first n programs have halted within n timesteps (and compute the lower bound that that implies on the constant)