11/15/2012, 06:34 PM
(This post was last modified: 11/16/2012, 02:27 AM by sheldonison.)
(11/14/2012, 08:30 PM)tommy1729 Wrote: Define the superfactorial as superfactorial(x) = super!(x) andHey Tommy,
super!(x+1) = factorial(super!(x)) = super!(x)!
Interesting question, I'll post more later. The fixed point=2, super! has a well defined schroder and inverse schroder function. I think it comes down to how big is \( \text{slog}(\log(x!)) \) and it seems like its not going to be that much bigger than \( \text{slog}(x) \). So \( \lim_{n \to \infty}\text{slog}(\text{super!}(n))-n \) should converge to a constant for integer values of n. I would expect a constant plus a 1-cyclic function real values of n.
updateThe \( \log(x!)\approx x\log(x) \), for large enough x, the log(x) multiplier term becomes arbitrarily accurate. \( \log(\log((x!)!)))\approx \log(x!\log(x!))\approx x\log(x)+\log(x)+\log(\log(x))\approx(x+1)\log(x)+\log(\log(x)) \).
The important thing is that slog((x+1)log(x)), if x is big enough, approaches arbitrarily close slog(x), so slog(super!(n))-n approaches a constant as n increases.
- Sheldon