alt Hacker NewsIndividual busy beavers BB(n) are finite natural numbers and thus quite computable. A related uncomputable number is the halting probability Omega of a universal prefix machine (whose programs form a prefix free set). By collecting enough halting programs to accumulate a probability of at least the first n bits of Omega (as a binary fraction), you will have determined all programs of length at most n that halt and thus also the busy beavers up to that size.
"A real number in which each decimal digit at position n is equal to the first digit of BB(n)."
Since you asserted that individual BB(n) numbers are computable, I think you will have no difficulty writing an algorithm that outputs that.