Speaking of equidistribution/normality, I wonder what's known about https://oeis.org/A018247 (the digits of a 10-adic fixed point of the squaring map). Has anyone read any of the references? REFERENCES W. W. R. Ball, Mathematical Recreations & Essays, N.Y. Macmillan Co, 1947. V. deGuerre and R. A. Fairbairn, Automorphic numbers, J. Rec. Math., 1 (No. 3, 1968), 173-179. M. Kraitchik, Sphinx, 1935, p. 1. I'm guessing that nobody has done a statistical analysis of the digits. Jim Propp On Wednesday, July 13, 2016, Neil Sloane <njasloane@gmail.com <javascript:_e(%7B%7D,'cvml','njasloane@gmail.com');>> wrote:
The sequence in question is https://oeis.org/A027878. Please also add any useful comments to that entry!
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Wed, Jul 13, 2016 at 11:45 AM, James Propp <jamespropp@gmail.com> wrote:
Yes, absolutely. That's why the initial and final digits behave so nicely. I should've mentioned that.
Jim
On Wednesday, July 13, 2016, Joerg Arndt <arndt@jjj.de> wrote:
Note that product(n>=1, 1-10^{-n}) is essentially Dedekind's eta function and Euler's pentagonal number theorem (to name just one) may be helpful with spotting patterns in the expansion.
Best regards, jj
* James Propp <jamespropp@gmail.com <javascript:;>> [Jul 13. 2016 08:00]:
One side-alley that I glanced down when I was preparing my current Mathematical Enchantments essay, but then resolutely marched myself past, involves the behavior of the decimal expansion of the product (10^1-1)(10^2-1)...(10^n-1); the initial and final digits are quite orderly, while the digits in the middle look random. Has anyone looked into this? It'd be especially interesting if one could establish some sort of phase transition.
I realize I'm using vocabulary loosely; there's no randomness involved, so we're not really doing stat mech. Still, some of the concepts of stat mech might apply.
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