User blog:Ikosarakt1/Inverse busy beaver function

Define inverse of busy beaver function, \(\Sigma^{-1}(n)\) as the minimum number of states needed to write n ones. This gives a sequence:

\(\Sigma^{-1}(1) = 1\)

\(\Sigma^{-1}(2) = 2\)

\(\Sigma^{-1}(3) = 2\)

\(\Sigma^{-1}(4) = 2\)

\(\Sigma^{-1}(5) = 3\)

\(\Sigma^{-1}(6) = 3\)

\(\Sigma^{-1}(7) = 4\)

\(\Sigma^{-1}(8) = 4\)

\(\Sigma^{-1}(9) = 4\)

\(\Sigma^{-1}(11) = 4\)

\(\Sigma^{-1}(12) = 4\)

\(\Sigma^{-1}(13) = 4\)

Note that \(\Sigma^{-1}(10)\) is not nesessarily 4, since it is possible that there is no TM with 4 states leaving exactly 10 ones. So, \(\Sigma^{-1}(n)\) may be not even strictly increasing!