User:Wythagoras/Rado's sigma function

\(\Sigma(6,3)\)
0 _ 1 r 1 0 1 2 l 0 0 2 1 l 2 1 _ _ l 0 1 1 2 r 1 1 2 1 l 1 2 _ 1 l 3 2 1 1 r 0 2 2 1 r 2 3 _ 1 r 3 3 1 _ l 3 3 2 1 r 4 4 _ 1 l halt 4 2 2 l 5 5 1 1 l 5 5 2 2 l 5 5 _ _ r 3 \(\Sigma(6,3) > 2^{374676384} > 10^{112788829}\)

All known bounds
\(\Sigma(1) = 1\), \(S(1) = 1\)

\(\Sigma(2) = 4\), \(S(2) = 6\)

\(\Sigma(3) = 6\), \(S(3) = 21\)

\(\Sigma(4) = 13\), \(S(4) = 107\)

\(\Sigma(5) ≥ 4098\), \(S(5) ≥ 47176870\)

\(\Sigma(6) > 3.514 \cdot 10^{18726}\), \(S(6) > 7.412 \cdot 10^{36534}\)

\(\Sigma(10) > 3 \uparrow^{3} 3\)

\(\Sigma(12) > 3 \uparrow^{4} 3\)

\(\Sigma(14) > 3 \uparrow^{5} 3\)

\(\Sigma(16) > 3 \uparrow^{6} 3\)

\(\Sigma(18) > 2 \uparrow^{2046} 4\)

\(\Sigma(19) > 2 \uparrow^{1.75 \cdot 10^{18726}} 4\)

\(\Sigma(25) > f_{\omega+1}(6413)\)

\(\Sigma(26) > f_{\omega+1}(5.25 \cdot 10^{18726})\)

3 colors
\(\Sigma(1,3) = 1\), \(S(1,3) = 1\)

\(\Sigma(2,3) = 9\), \(S(2,3) = 38\)

\(\Sigma(3,3) = 374676383\), \(S(3,3) = 119112334170342540\)

\(\Sigma(4,3) > 1.383 \cdot 10^{7036}\), \(S(4,3) > 1.025 \cdot 10^{14072}\)

\(\Sigma(6,3) > 2^{374676384}\)

\(\Sigma(7,3) > 2^{2^{2^{2^{2^{2^{33}}}}}}\)

\(\Sigma(48,3) > f_{\varepsilon_0}(374676379)\)

4 colors
\(\Sigma(1,4) = 1\), \(S(1,4) = 1\)

\(\Sigma(2,4) = 2050\), \(S(2,4) = 3932964\)

\(\Sigma(3,4) > 3.7 \cdot 10^{6518}\), \(S(3,4) > 5.2 \cdot 10^{13036}\)

\(\Sigma(6,4) > 2^{2^{256}}\)

\(\Sigma(17,4) > f_{\omega+1}(2046)\)

5 colors
\(\Sigma(1,5) = 1\), \(S(1,5) = 1\)

\(\Sigma(2,5) > 1.7 \cdot 10^{352}\), \(S(2,5) > 1.9 \cdot 10^{704}\)

\(\Sigma(5,5) > 2 \uparrow\uparrow 8 - 2\)

\(\Sigma(6,5) > 10 \uparrow\uparrow (1.7*10^{352})\)

6 colors
\(\Sigma(1,6) = 1\), \(S(1,6) = 1\)

\(\Sigma(2,6) > 1.9 \cdot 10^{4933}\), \(S(2,6) > 2.4 \cdot 10^{9866}\)

\(\Sigma(6,6) > (2 \uparrow\uparrow)^6(2^{258}-3)-2\)

Subpages
Progress on the 42 Hardly Non-Regular TMs Multi-Headed TMs