A129664 Numerators 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.

0, 1, 5, 53, 25619, 73767966817, 388826530522004941794623, 226073434564505101198889656344981223287273794070917, 302470760179203901700754265690364240921018701177125350099844323581396873793766696160680079412655525143887

Offset

0,3

References

Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 176 and 292

Links

Table of n, a(n) for n=0..8.

Formula

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.

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 + ...

Closed form: L(3, chi3) = 4 Pi^3/(81 sqrt(3)).

Example

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, ...

Mathematica

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]; Numerator[ FoldList[Plus, 0, 1/e] ]

Crossrefs

Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.

Cf. A129658, A129659, A129660, A129661, A129662, A129663, A129665.

Sequence in context: A236529 A083757 A093674 * A275499 A275396 A275261

Adjacent sequences: A129661 A129662 A129663 * A129665 A129666 A129667

Keyword

nonn,frac,easy

Author

Stuart Clary, Apr 30 2007

Status

approved