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A114362 Numerator of zeta(4n)/zeta(2n)^2 (with a(0)=2 instead of -2). 9
2, 2, 6, 691, 7234, 523833, 3545461365, 3392780147, 15418642082434, 26315271553053477373, 261082718496449122051, 2530297234481911294093, 39265823582984723803743892829, 61628132164268458257532691681 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

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

LINKS

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.

FORMULA

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

PROG

(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))

CROSSREFS

Cf. A027641, A114363, A348829, A348830.

Sequence in context: A304564 A181265 A093909 * A007338 A164325 A198880

Adjacent sequences:  A114359 A114360 A114361 * A114363 A114364 A114365

KEYWORD

frac,nonn

AUTHOR

Benoit Cloitre, Feb 09 2006; corrected Feb 22 2006

STATUS

approved

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Last modified May 25 19:19 EDT 2022. Contains 354071 sequences. (Running on oeis4.)