A171728 Numbers k which establish records for floor(log(log(log(2^k)))).

2, 3, 4, 22, 2335, 762451795, 742762245454927736743542, 41133018324375596439235122590123953570787986963829981156569123587

Offset: 1

Author

Jonathan Vos Post, Dec 16 2009

Keywords

• easy
• nonn

Comments

Morris writes: E. Thorp introduced the following card shuffling model. Suppose the number of cards n is even. Cut the deck into two equal piles. Drop the first card from the left pile or from the right pile according to the outcome of a fair coin flip. Then drop from the other pile. Continue this way until both piles are empty. We show that if n is a power of 2 then the mixing time of the Thorp shuffle is O(log^3 n). Previously, the best known bound was O(log^4 n).

This sequence seems to be unrelated to the Thorp shuffle in which the bound is log^3 x = (log x)^3 rather than log log log x. - Charles R Greathouse IV, Sep 04 2015

Examples

			a(1) = 2 because log(log(log(2^2))) ~ -1.1189142050548055457 whose floor is -2.
a(2) = 3 because log(log(log(2^3))) ~ -0.31183902548187902095 whose floor is -1.
		

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..11
• Ben Morris, Improved mixing time bounds for the Thorp shuffle, arXiv:0912.2759 [math.PR], Dec 14, 2009.

Crossrefs

Cf. A000079, A000195.

Programs

Mathematica

a[n_] := Ceiling[Exp[Exp[n - 3] - Log@ Log@ 2]]; Array[a, 11] (* Robert G. Wilson v, Feb 05 2013 *)

PARI

a(n)=ceil(exp(exp(n-3))/log(2)) \\ Charles R Greathouse IV, Dec 20 2011

Formula

a(n) = Min(n such that floor(log(log(log(2^n)))) > floor(log(log(log(2^(n-1)))))).

a(n) = ceiling(exp(exp(n-3)-log(log(2)))). - R. J. Mathar, Mar 31 2010

Extensions

Two more terms from R. J. Mathar, Mar 31 2010