Busy beaver function

Rado's sigma function (or busy beaver function) $$\Sigma(n)$$ is equal to the maximum number of 1s that can be written with an n-state, 2-color Turing machine before halting. Turing machines that produce these numbers are called busy beavers.

The only known values for $$\Sigma(n)$$ are $$\Sigma(1) = 1, \Sigma(2) = 4, \Sigma(3) = 6,$$ and $$\Sigma(4) = 13$$. The next two numbers are very large, and the known lower bounds are $$\Sigma(5) \geq 4098$$ and $$\Sigma(6) \geq 3.514 \cdot 10^{18276}$$.

Rado's sigma function is uncomputable &mdash; determining whether a given Turing machine is a busy beaver is an undecidable problem. It grows extremely fast; Rado showed that it grows faster than any computable function.