The busy beaver numbers form an uncomputable sequence.

For BB(5) the proof of its value is an indirect computation. The verification process involved both computation (running many machines) and proofs (showing others run forever or halt earlier). The exhaustiveness of crowdsourced proofs was a tour de force.

Isn't it rather that the Busy Beaver function is uncomputable, particular values can be calculated - although anything great than BB(5) is quite a challenge!

https://scottaaronson.blog/?p=8972

They are at the very boundary of what is computable!