This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A160755 #26 Feb 16 2025 08:33:10
%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,
%T A160755 26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,
%U A160755 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 Adding the series -1+sqrt(2)-3^(1/3)+4^(1/4)..., according to this sequence, 10 billion terms must be added to arrive at 11 accurate digits of the MRB constant.
%D A160755 Steven R. Finch, Mathematical Constants, Cambridge, 2003, p. 450. ISBN 0521818052.
%H A160755 Henri Cohen, Fernando Rodriguez Villegas and Don Zagier, <a href="https://projecteuclid.org/journals/experimental-mathematics/volume-9/issue-1/Convergence-acceleration-of-alternating-series/em/1046889587.full">Convergence Acceleration of Alternating Series</a>, Experimental Mathematics, 9:1 (2000).
%H A160755 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/MRBConstant.html">MRB Constant</a>.
%H A160755 Wikipedia, <a href="http://en.wikipedia.org/wiki/Mathematical_constant">Mathematical constant</a>
%e A160755 For n=1, a(n)=1 because after 10^1 partial sums of -1+sqrt(2)-3^(1/3)+4^(1/4)... you get one accurate digit of the MRB constant.
%e A160755 For n=2, a(n)=2 because after 10^2 partial sums you get 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 Cf. A037077 (the MRB constant).
%K A160755 nonn,base,less
%O A160755 1,2
%A A160755 _Marvin Ray Burns_, May 25 2009
%E A160755 Corrections from _Marvin Ray Burns_, Jun 05 2009
%E A160755 Link to Wikipedia replaced by up-to-date version; keyword:less added _R. J. Mathar_, Aug 04 2010
%E A160755 Corrections by _Marvin Ray Burns_, Aug 21 2010, Jul 15 2012