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A114363 Denominator of zeta(4n)/zeta(2n)^2.

%I A114363 #50 Nov 06 2024 15:15:51
%S A114363 1,5,7,715,7293,524875,3547206349,3393195750,15419113345821,
%T A114363 26315472459271727875,261083216622451556697,2530298441183206558150,
%U A114363 39265828264113994596230058165,61628134000978439089402342590
%N A114363 Denominator of zeta(4n)/zeta(2n)^2.
%C A114363 zeta(4n)/zeta(2n)^2 is a rational value expressible in term of Bernoulli's numbers (A027641).
%H A114363 Seiichi Manyama, <a href="/A114363/b114363.txt">Table of n, a(n) for n = 0..158</a>
%F A114363 For n > 0, Product_{p primes} (p^{2n}-1)/(p^{2n}+1) = zeta(4n)/zeta(2n)^2.
%F A114363 For n > 0, a(n) = Denominator((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_, Feb 12 2022
%e A114363 -2/1, 2/5, 6/7, 691/715, 7234/7293, 523833/524875, 3545461365/3547206349, ...
%t A114363 a[ n_] := If[ n < 0, 0, Denominator[ Zeta[4*n] / Zeta[2*n]^2 ]] (* _Michael Somos_, Jan 27 2012 *)
%o A114363 (PARI) z(n)=bernfrac(2*n)*(-1)^(n - 1)*2^(2*n-1)/(2*n)!;
%o A114363 a(n)=if(n<1,1,denominator(z(2*n)/z(n)^2))
%Y A114363 Cf. A027641, A027642, A114362 (numerators), A348829, A348830.
%K A114363 frac,nonn
%O A114363 0,2
%A A114363 _Benoit Cloitre_, Feb 09 2006; corrected Feb 22 2006