%I A160755
%S A160755 1,2,3,4,5,6,7,8,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,
%T A160755 27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49
%N A160755 Number of correct digits of the MRB constant derived from the sequence 
               of partial sums up to m=10^n terms as defined by S[n]= Sum[(-1)^k*(k^(1/
               k)-1),{k,m}]
%C A160755 If one would fail to use acceleration methods, then according to this 
               sequence 10^49 terms must be computed and added (processed) to arrive 
               at 50 digits of the MRB Constant.
%C A160755 So at 10^10 Hertz and with 2^16 terms processed per Hertz, it would take 
               176,606,354,890,046,296,296,296,296,296 days to compute 50 digits 
               of the MRB Constant.
%C A160755 Compute days as follows:
%C A160755 The 50th term is 49
%C A160755 and
%C A160755 rate = 10^10*2^16
%C A160755 days = 10^49/rate/3600/24.
%D A160755 S. R. Finch, Mathematical Constants, Cambridge, 2003, p. 450. ISBN 0521818052.
%D A160755 Henri Cohen, Fernando Rodriguez Villegas, and Don Zagier, "Convergence 
               Acceleration of Alternating Series", Experimental Mathematics, 9:1 
               (2000).
%H A160755 Weisstein, Eric W. "MRB Constant"; <a href="http://mathworld.wolfram.com/
               MRBConstant.html">http://mathworld.wolfram.com/MRBConstant.html</
               a>
%H A160755 Wikipedia contributors, 'Mathematical constant', Wikipedia, The Free 
               Encyclopedia, 23 May 2009, 18:49 UTC, <a href="http://en.wikipedia.org/
               w/index.php?title=Mathematical_constant&oldid=291856790#Table_of_selected_mathematical_constants">
               Table of selected mathematical constants</a> [accessed 25 May 2009]
%F A160755 Where A004709 drops an 8 add two 8's.
%e A160755 After 10^1 partial sums you get one accurate digit; 10^2 partial sums 
               = two accurate digits and so on.
%t A160755 m = NSum[(-1)^n*(n^(1/n) - 1), {n, Infinity}, Method -> "AlternatingSigns", 
               WorkingPrecision -> 1000]; Table[-Floor[Log[10, Abs[m - NSum[(-1)^n*(n^(1/
               n) - 1), {n, 10^a}, Method ->"AlternatingSigns", WorkingPrecision 
               -> 1000]]]], {a,1, 50}]
%Y A160755 A037077 gives the sequence of digits of the MRB constant.
%Y A160755 Sequence in context: A108922 A102670 A079631 this_sequence A017873 A128557 
               A103303
%Y A160755 Adjacent sequences: A160752 A160753 A160754 this_sequence A160756 A160757 
               A160758
%K A160755 nonn
%O A160755 1,2
%A A160755 Marvin Ray Burns (bmmmburns(AT)sbcglobal.net), May 25 2009
%E A160755 Corrections from Marvin Ray Burns (bmmmburns(AT)sbcglobal.net), Jun 05 
               2009