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A129661 Denominators of the Engel partial sums for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.

%I A129661 #4 Dec 09 2016 13:21:53
%S A129661 1,2,4,8,112,10528,3916416,453977888,5984725643520,24757413551258752,
%T A129661 36544913291284069002240,3209228105587401803500707840,
%U A129661 206085396642453387914503205007360
%N A129661 Denominators of the Engel partial sums for L(3, chi3), where L(s, chi3) is the Dirichlet L-function for the non-principal character modulo 3.
%D A129661 Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292
%F A129661 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 A129661 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 A129661 Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).
%e A129661 L(3, chi3) = 0.8840238117500798567430579168710118077... = 1/2 + 1/(2*2) + 1/(2*2*2) + 1/(2*2*2*14) + 1/(2*2*2*14*94) + ..., the partial sums of which are 0, 1/2, 3/4, 7/8, 99/112, 9307/10528, ...
%t A129661 nmax = 100; prec = 2000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; e = First@Transpose@NestList[{Ceiling[1/(#[[1]] #[[2]] - 1)], #[[1]] #[[2]] - 1}&, {Ceiling[1/c], c}, nmax - 1]; Denominator[ FoldList[Plus, 0, 1/Drop[ FoldList[Times, 1, e], 1 ] ] ]
%Y A129661 Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
%Y A129661 Cf. A129658, A129659, A129660, A129662, A129663, A129664, A129665.
%K A129661 nonn,frac,easy
%O A129661 0,2
%A A129661 _Stuart Clary_, Apr 30 2007