cp's OEIS Frontend

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

%I A171728 #33 May 10 2025 09:15:49
%S A171728 2,3,4,22,2335,762451795,742762245454927736743542,
%T A171728 41133018324375596439235122590123953570787986963829981156569123587
%N A171728 Numbers k which establish records for floor(log(log(log(2^k)))).
%C A171728 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).
%C A171728 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
%H A171728 Robert G. Wilson v, <a href="/A171728/b171728.txt">Table of n, a(n) for n = 1..11</a>
%H A171728 Ben Morris, <a href="http://arxiv.org/abs/0912.2759">Improved mixing time bounds for the Thorp shuffle</a>, arXiv:0912.2759 [math.PR], Dec 14, 2009.
%F A171728 a(n) = Min(n such that floor(log(log(log(2^n)))) > floor(log(log(log(2^(n-1)))))).
%F A171728 a(n) = ceiling(exp(exp(n-3)-log(log(2)))). - _R. J. Mathar_, Mar 31 2010
%e A171728 a(1) = 2 because log(log(log(2^2))) ~ -1.1189142050548055457 whose floor is -2.
%e A171728 a(2) = 3 because log(log(log(2^3))) ~ -0.31183902548187902095 whose floor is -1.
%t A171728 a[n_] := Ceiling[Exp[Exp[n - 3] - Log@ Log@ 2]]; Array[a, 11] (* _Robert G. Wilson v_, Feb 05 2013 *)
%o A171728 (PARI) a(n)=ceil(exp(exp(n-3))/log(2)) \\ _Charles R Greathouse IV_, Dec 20 2011
%Y A171728 Cf. A000079, A000195.
%K A171728 easy,nonn
%O A171728 1,1
%A A171728 _Jonathan Vos Post_, Dec 16 2009
%E A171728 Two more terms from _R. J. Mathar_, Mar 31 2010