The maximum number of ones that a halting turing machine with $n$ states and $2$ symbols can write down on an initially empty tape , if the ones must form a single block, is usually denoted with $\sigma(n)$. The maximum number of steps should also differ from $S(n)$
Does anyone know a site with a survey of the current champions in this category ?
For $n\le 3$, the values do not differ because there are busy beavers forming a single block. But for $4$ states, the champion I found online does not produce a single block, neither does the current busy beaver I found with $5$ states.