I should explain a little more. If, as your friend did on his web page, puts a decimal point after the first digit of 2^n, and then looks at the ordering (i.e. sorts them), this is the same order as if you sorted the log base 10 of 2^n modulo 1 (i.e. subtract the integral part). The Weyl equidistribution theorem says that for any interval [a,b] in [0,1] that the ratio of the number of n alpha mod 1, for n=1, ..., N that land in that interval to N converges to b-a as N --> infinity. Since we're taking logarithms base 10, we're comparing that with the sequence n/10 mod 1. For a quanitative version look at the Erdos-Turan inequality: https://en.wikipedia.org/wiki/Erdős–Turán_inequality On Sun, Jul 28, 2019 at 5:08 PM Simon Plouffe <simon.plouffe@gmail.com> wrote:
Hello ,
is it what you mean ? See sequences : http://oeis.org/A158624 and : https://oeis.org/A023415 These are the limit values of 5^n and 2^n backward.
Simon Plouffe
and sequence
Le dim. 28 juil. 2019 à 22:58, Victor Miller <victorsmiller@gmail.com> a écrit :
What your friend observed is essentially a consequence of Weyl’s theorem: Let alpha be irrational, then the sequence n alpha mod 1 is dense in the interval [0,1]. Since log 2/log 10 is irrational, this applies.
Victor
On Fri, Jul 26, 2019 at 09:12 Colin Wright <maths@solipsys.co.uk> wrote:
Here's a small thing that a friend of mine discovered:
https://www.solipsys.co.uk/new/PowersOfTwoInLexOrder.html?sg26mf
Surely this has been seen before, but I can't find any kinds of references to it. If anyone can suggest some references I'd be grateful.
Thanks.
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