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Search: id:A129660
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| A129660 |
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Numerators 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. |
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16
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| 0, 1, 3, 7, 99, 9307, 3462205, 401327263, 5290639975663, 21886143096656843, 32306573547837099089161, 2837034062676862693613762377, 182184397885888753164448171682621
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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Leonhard Euler, ``Introductio in Analysin Infinitorum'', First Part, Articles 176 and 292
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FORMULA
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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))
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EXAMPLE
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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, ...
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MATHEMATICA
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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]; Numerator[ FoldList[Plus, 0, 1/Drop[ FoldList[Times, 1, e], 1 ] ] ]
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CROSSREFS
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Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
Cf. A129658, A129659, A129661, A129662, A129663, A129664, A129665.
Sequence in context: A062592 A074349 A159310 this_sequence A158467 A028414 A014014
Adjacent sequences: A129657 A129658 A129659 this_sequence A129661 A129662 A129663
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KEYWORD
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nonn,frac,easy
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AUTHOR
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Stuart Clary (clary(AT)uakron.edu), Apr 30, 2007
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