User:Wythagoras/Rado's sigma function/BB(13,3)

\(\Sigma(13,3) > f_{\omega}(8)\)

0 _ 1 r bb1 0 1 2 l bb1 0 2 1 l 8 bb1 _ 2 l 0 bb1 1 2 r bb1 bb1 2 1 l bb1 0a _ 1 r 0a 0a 1 _ l 0a 0a 2 1 r 1 1 1 2 l 3 1 2 2 r 2 1 _ 1 r halt 2 _ 1 r halt 2 2 2 r 2 2 1 2 l 8 3 2 1 l 3 3 1 1 l 3 3 _ 1 l 0a 4 2 1 r 5 4 1 1 l 4 4 _ 2 r 6 5 2 2 l 9 5 1 2 l 7 5 _ _ l 7 6 1 1 r 6 6 2 2 * 4 7 1 1 l 7 7 2 2 l 7 7 _ 1 l 0a 8 2 2 l 8 8 1 1 r 4 8 _ 2 l 7 9 1 1 l 9 9 2 _ l 9 9 _ 2 r 10 10 1 1 r 10 10 2 2 * 4 10 _ 2 r 10

Explanation
The first part is the machine for \(\Sigma(2,3)\). It outputs: 222222212      ^

It uses groups, the first group duplicates, the second group is comparable to \(2^n\), the third group is comparable to \(2\uparrow\uparrow n\), the fourth group is comparable to \(2\uparrow\uparrow\uparrow n\), etc. It uses LittlePeng9's duplication machine.