A171728 - OEIS (original) (raw)

A171728

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

1

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

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

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

EXAMPLE

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.

MATHEMATICA

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