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

1, 5, 7, 715, 7293, 524875, 3547206349, 3393195750, 15419113345821, 26315472459271727875, 261083216622451556697, 2530298441183206558150, 39265828264113994596230058165, 61628134000978439089402342590

Offset: 0

Author

Benoit Cloitre, Feb 09 2006; corrected Feb 22 2006

Keywords

• frac
• nonn

Comments

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

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..158

Crossrefs

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

Programs

Mathematica

a[ n_] := If[ n < 0, 0, Denominator[ Zeta[4*n] / Zeta[2*n]^2 ]] (* Michael Somos, Jan 27 2012 *)

PARI

z(n)=bernfrac(2*n)*(-1)^(n - 1)*2^(2*n-1)/(2*n)!;
a(n)=if(n<1,1,denominator(z(2*n)/z(n)^2))

Formula

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

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