User blog comment:Ikosarakt1/Inverse busy beaver function/@comment-1605058-20140311153328

We can also give an upper bound on \(\Sigma^{-1}(n)\) - n can be easily written by n-state machine, so \(\Sigma^{-1}(n)\leq n\). We can't give easily computable, and non-trivial, lower bound, but we can give an upper bound on the least n such that, for given k, \(\Sigma^{-1}(n)>k\). Namely, we have only (4(k+1))^2k machines with k states, so we have at most (4(k+1))^2k numbers x such that \(\Sigma^{-1}(x)\leq k\). Thus least n in question is at most (4(k+1))^2k+1, which we can easily improve and know that n<(4(k+1))^2k.