 +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Iterations of ln(mod(ln(x^(1/x))) and number with property Left an=exp (-(a^(n+1)) (/showthread.php?tid=139) Iterations of ln(mod(ln(x^(1/x))) and number with property Left an=exp (-(a^(n+1)) - Ivars - 03/30/2008 I have been playing with iterations of And of course I tried and is not very interesting, it converges to 2 values depending on integer iteration number. Here are picture of 200 iterations: [attachment=283] is more interesting. This iteration converges to 4 values for each x, cyclicaly,and their dependance on n is shifting depending the region x is in the interval ]0:1[. The convergence values are : Here is what happens: [attachment=284] [attachment=287] When resolution is increased (step decreased) more and more shifts in phase happen in the region which seems to converge to approximately 0,6529204.... [attachment=285] [attachment=286] This number has properties: So its 2nd selfroot is , while its reciprocal 1,531580266.. is 3rd superroot of e. It has also following properties, at least approximately numerically, so it might be wrong, but interesting: if we denote it , than: So: ........ .......... if this is so, what happens if instead of integer n we take x, so that: maybe: I hope I used [4Left] correctly. So: and If accuracy is not enough, so it is numerically only 3-4 digits, perhaps going for n-th superroot of e would improve situation? Probably this is old knowledge. Ivars