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A129665 Denominators of the greedy Egyptian partial sums for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

%I A129665 #4 Dec 09 2016 13:19:12
%S A129665 1,2,6,60,28980,83445678540,439837168811386168898460,
%T A129665 255732290872293553071304874994266857210112979247740,
%U A129665 342152277075444487917411768449441971426262505651282338530700909926424044202121143490579209389129867953540
%N A129665 Denominators of the greedy Egyptian partial sums for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
%D A129665 Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
%F A129665 chi3(k) = Kronecker(-3, k); chi3(k) is 0, 1, -1 when k reduced modulo 3 is 0, 1, 2, respectively; chi3 is A049347 shifted.
%F A129665 Series: L(3, chi3) = Sum_{k=1..infinity} chi3(k) k^{-3} = 1 - 1/2^3 + 1/4^3 - 1/5^3 + 1/7^3 - 1/8^3 + 1/10^3 - 1/11^3 + ...
%F A129665 Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
%e A129665 L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/2 + 1/3 + 1/20 + 1/1449 + 1/2879423 + ..., the partial sums of which are 0, 1/2, 5/6, 53/60, 25619/28980, 73767966817/83445678540, ...
%t A129665 nmax = 12; prec = 2000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; e = First@Transpose@NestList[{Ceiling[1/(#[[2]] - 1/#[[1]])], #[[2]] - 1/#[[1]]}&, {Ceiling[1/c], c}, nmax - 1]; Denominator[ FoldList[Plus, 0, 1/e] ]
%Y A129665 Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
%Y A129665 Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129664.
%K A129665 nonn,frac,easy
%O A129665 0,2
%A A129665 _Stuart Clary_, Apr 30 2007