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 My favorite integer sequence sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 08/04/2014, 11:49 PM (This post was last modified: 08/05/2014, 01:41 AM by sheldonison.) (08/03/2014, 11:38 PM)tommy1729 Wrote: $f^{o n}(x)$ grows like $exp(a^{2^n - 1} ln(x)^{2^n} + C)$. Much much slower then exp or even its half-iterate. And that for a function that is similarly defined. (second derivative = Original => second difference = Original ) Thats what I meant by counterintuitive. I noticed the resemblance of the plots with the fake half-iterate exp too. For the zero's Im convinced they are all real. I made some numerical approximations, and for big enough values of x, using Jay's Taylor series, I get that $\ln(f(x)) \approx \frac{(\ln(x))^2}{2\ln(2)}$, which match the equations in Gottfried's link. If ln(x)=1E100, then ln(f(x))~=7.213E199, ln(f(f(x)))~=3.753E399 So then $\ln(f(f(x))) \approx (\ln(x))^4$. And yeah, that's growing much slower than $\exp^{0.5}(\exp^{0.5}(x))=\exp(x) \;\; \exp(x) \gt \exp((\ln(x))^4)$, if x is big enough. This is despite many similarities in the complex plane graphs of f(x) and f(f(x)) as compared with the entire asymptotic to $\exp^{0.5}(x)$. - Sheldon « Next Oldest | Next Newest »

 Messages In This Thread My favorite integer sequence - by tommy1729 - 07/27/2014, 08:44 PM RE: My favorite integer sequence - by jaydfox - 07/31/2014, 01:51 AM RE: My favorite integer sequence - by jaydfox - 08/01/2014, 01:53 AM RE: My favorite integer sequence - by Gottfried - 08/01/2014, 03:25 PM RE: My favorite integer sequence - by jaydfox - 08/01/2014, 05:04 PM RE: My favorite integer sequence - by sheldonison - 08/01/2014, 11:36 PM RE: My favorite integer sequence - by jaydfox - 08/01/2014, 11:57 PM RE: My favorite integer sequence - by jaydfox - 08/05/2014, 04:49 PM RE: My favorite integer sequence - by jaydfox - 08/05/2014, 05:57 PM RE: My favorite integer sequence - by Gottfried - 08/06/2014, 04:38 PM RE: My favorite integer sequence - by jaydfox - 08/07/2014, 01:19 AM RE: My favorite integer sequence - by jaydfox - 08/07/2014, 01:34 AM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 09:42 AM RE: My favorite integer sequence - by Gottfried - 08/09/2014, 02:27 PM RE: My favorite integer sequence - by tommy1729 - 09/09/2014, 12:55 AM RE: My favorite integer sequence - by jaydfox - 08/08/2014, 12:55 AM RE: My favorite integer sequence - by Gottfried - 08/08/2014, 02:27 AM RE: My favorite integer sequence - by jaydfox - 09/09/2014, 07:43 PM RE: My favorite integer sequence - by jaydfox - 09/09/2014, 09:45 PM RE: My favorite integer sequence - by jaydfox - 08/02/2014, 12:08 AM RE: My favorite integer sequence - by tommy1729 - 08/03/2014, 11:38 PM RE: My favorite integer sequence - by sheldonison - 08/04/2014, 11:49 PM RE: My favorite integer sequence - by jaydfox - 09/16/2014, 05:32 AM RE: My favorite integer sequence - by Gottfried - 09/17/2014, 07:39 PM RE: My favorite integer sequence - by jaydfox - 10/02/2014, 10:53 PM RE: My favorite integer sequence - by Gottfried - 08/03/2014, 03:32 PM RE: My favorite integer sequence - by tommy1729 - 08/03/2014, 11:44 PM RE: My favorite integer sequence - by sheldonison - 08/02/2014, 05:48 AM RE: My favorite integer sequence - by tommy1729 - 09/10/2014, 08:57 PM RE: My favorite integer sequence - by Gottfried - 08/02/2014, 07:43 PM RE: My favorite integer sequence - by Gottfried - 08/02/2014, 09:29 PM RE: My favorite integer sequence - by Gottfried - 08/02/2014, 09:36 PM RE: My favorite integer sequence - by tommy1729 - 08/05/2014, 11:16 PM RE: My favorite integer sequence - by tommy1729 - 08/08/2014, 11:02 PM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 07:02 PM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 10:51 PM RE: My favorite integer sequence - by sheldonison - 08/11/2014, 04:51 PM RE: My favorite integer sequence - by jaydfox - 08/11/2014, 05:19 PM RE: My favorite integer sequence - by jaydfox - 08/19/2014, 01:36 AM RE: My favorite integer sequence - by jaydfox - 08/19/2014, 02:05 AM RE: My favorite integer sequence - by jaydfox - 08/19/2014, 05:31 PM RE: My favorite integer sequence - by sheldonison - 08/19/2014, 07:56 PM RE: My favorite integer sequence - by jaydfox - 08/20/2014, 07:42 AM RE: My favorite integer sequence - by sheldonison - 08/20/2014, 02:11 PM RE: My favorite integer sequence - by jaydfox - 08/20/2014, 07:57 PM RE: My favorite integer sequence - by jaydfox - 08/21/2014, 01:15 AM RE: My favorite integer sequence - by jaydfox - 08/21/2014, 05:25 AM RE: My favorite integer sequence - by jaydfox - 08/22/2014, 05:39 PM RE: My favorite integer sequence - by jaydfox - 09/11/2014, 01:33 AM RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 09:16 PM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 10:19 PM RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 10:52 PM RE: My favorite integer sequence - by jaydfox - 08/09/2014, 11:46 PM RE: My favorite integer sequence - by tommy1729 - 08/09/2014, 11:10 PM RE: My favorite integer sequence - by jaydfox - 08/10/2014, 12:30 AM RE: My favorite integer sequence - by tommy1729 - 08/11/2014, 12:17 PM RE: My favorite integer sequence - by Gottfried - 08/22/2014, 12:30 AM Amazing variant - by tommy1729 - 08/26/2014, 08:57 PM RE: My favorite integer sequence - by tommy1729 - 09/01/2014, 10:37 PM RE: My favorite integer sequence - by tommy1729 - 10/02/2014, 11:24 PM RE: My favorite integer sequence - by jaydfox - 10/02/2014, 11:29 PM RE: My favorite integer sequence - by tommy1729 - 02/10/2015, 12:15 AM RE: My favorite integer sequence - by tommy1729 - 02/15/2015, 05:19 PM RE: My favorite integer sequence - by tommy1729 - 10/07/2015, 08:22 AM RE: My favorite integer sequence - by tommy1729 - 10/07/2015, 09:10 PM RE: My favorite integer sequence - by tommy1729 - 03/13/2016, 12:31 AM

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