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A114362
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Numerator of zeta(4n)/zeta(2n)^2 (with a(0)=2 instead of -2).
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9
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2, 2, 6, 691, 7234, 523833, 3545461365, 3392780147, 15418642082434, 26315271553053477373, 261082718496449122051, 2530297234481911294093, 39265823582984723803743892829, 61628132164268458257532691681
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OFFSET
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0,1
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COMMENTS
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zeta(4n)/zeta(2n)^2 is a rational value expressible in term of Bernoulli's numbers (A027641).
Conjecture: if an integer n > 1 is odd, then zeta(2n)/zeta(n)^2 is irrational. Cf. W. Kohnen (link) and my conjecture in A348829. - Thomas Ordowski, Jan 05 2022
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LINKS
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Seiichi Manyama, Table of n, a(n) for n = 0..158
Winfried Kohnen, Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 99, No. 3 (1989), pp. 231-233.
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FORMULA
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Product_{p primes} (p^{2n}-1)/(p^{2n}+1) = zeta(4n)/zeta(2n)^2.
For n > 0, a(n) = Numerator((D(n) - N(n)) / (D(n) + N(n))), where N(n) = A348829(n) and D(n) = A348830(n). See my comments and formulas in A348829. - Thomas Ordowski, Jan 05 2022
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PROG
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(PARI) z(n)=bernfrac(2*n)*(-1)^(n - 1)*2^(2*n-1)/(2*n)!;
a(n)=if(n<1, 2, numerator(z(2*n)/z(n)^2))
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CROSSREFS
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Cf. A027641, A114363, A348829, A348830.
Sequence in context: A304564 A181265 A093909 * A007338 A164325 A198880
Adjacent sequences: A114359 A114360 A114361 * A114363 A114364 A114365
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KEYWORD
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frac,nonn
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AUTHOR
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Benoit Cloitre, Feb 09 2006; corrected Feb 22 2006
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STATUS
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approved
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