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Search: id:A129663
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| A129663 |
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Denominators of the Pierce 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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| 1, 1, 8, 26, 1664, 106496, 370126848, 7279690096640, 4045738169062195200, 597704977138451388530688000, 111845949979901797334235660288000, 1194765595895193218918930427630975811584000
(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... = 1/1 - 1/(1*8) + 1/(1*8*13) - 1/(1*8*13*16) + 1/(1*8*13*16*64) - ..., the partial sums of which are 0, 1, 7/8, 23/26, 1471/1664, 94145/106496, ...
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MATHEMATICA
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nmax = 100; prec = 3000 (* Adjust the precision depending on nmax. *); c = N[ 4 Pi^3/(81 Sqrt[3]), prec]; p = First@Transpose@NestList[{Floor[ 1/(1 - #[[1]] #[[2]]) ], 1 - #[[1]] #[[2]]}&, {Floor[1/c], c}, nmax - 1]; p = Drop[ FoldList[Times, 1, p], 1 ]; Denominator[ FoldList[ Plus, 0, (-1)^Range[0, Length[p] - 1]/p ] ]
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CROSSREFS
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Cf. A129404, A129405, A129406, A129407, A129408, A129409, A129410, A129411.
Cf. A129658, A129659, A129660, A129661, A129662, A129664, A129665.
Sequence in context: A112645 A089064 A000810 this_sequence A112646 A084825 A119522
Adjacent sequences: A129660 A129661 A129662 this_sequence A129664 A129665 A129666
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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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