Busy beaver function

Rado's sigma function $$\Sigma(n)$$ is equal to the maximum number of 1s that can be written with an n-state, 2-color Turing machine. 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 known lower bounds for the next two numbers are $$\Sigma(5) \geq 4098$$ and $$\Sigma(6) \geq 1.29 \cdot 10^{85}$$, as described by H. Marxen.