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Denominator of zeta(4n)/zeta(2n)^2.
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%I #50 Nov 06 2024 15:15:51

%S 1,5,7,715,7293,524875,3547206349,3393195750,15419113345821,

%T 26315472459271727875,261083216622451556697,2530298441183206558150,

%U 39265828264113994596230058165,61628134000978439089402342590

%N Denominator of zeta(4n)/zeta(2n)^2.

%C zeta(4n)/zeta(2n)^2 is a rational value expressible in term of Bernoulli's numbers (A027641).

%H Seiichi Manyama, <a href="/A114363/b114363.txt">Table of n, a(n) for n = 0..158</a>

%F For n > 0, Product_{p primes} (p^{2n}-1)/(p^{2n}+1) = zeta(4n)/zeta(2n)^2.

%F 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 -2/1, 2/5, 6/7, 691/715, 7234/7293, 523833/524875, 3545461365/3547206349, ...

%t a[ n_] := If[ n < 0, 0, Denominator[ Zeta[4*n] / Zeta[2*n]^2 ]] (* _Michael Somos_, Jan 27 2012 *)

%o (PARI) z(n)=bernfrac(2*n)*(-1)^(n - 1)*2^(2*n-1)/(2*n)!;

%o a(n)=if(n<1,1,denominator(z(2*n)/z(n)^2))

%Y Cf. A027641, A027642, A114362 (numerators), A348829, A348830.

%K frac,nonn

%O 0,2

%A _Benoit Cloitre_, Feb 09 2006; corrected Feb 22 2006