Busy Beaver function

Summary

The Busy Beaver function $\Sigma(n)$ is the largest number of $1$s left by a halting Turing machine with $n$ states and two symbols. The machine starts on a blank tape. The related function $S(n)$ is the largest number of steps such a machine can take before it halts. Both functions are well defined, because there are only finitely many such machines. They grow faster than every computable function, so they are not computable, which gives a concrete example of a non-computable function.