How many odds could a Collatz counter-example cycle have, relative to its length?

[Collag3n]

I see you often use this upper bound \cfrac{3^x\,(x/2)}{2^k-3^x}, but there is a better one: \cfrac{3^x\,(x/3)}{2^k-3^x}
It comes from this https://math.meta.stackexchange.com/revisions/4669/655 where you have \frac{2^S}{3^N}<\frac{3a_0+N}{3a_0} leading to a_0<\frac{N3^{N-1}}{2^S-3^N}