This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A231273 #142 Feb 28 2022 01:52:35
%S A231273 1,1,1,691,3617,174611,236364091,3392780147,7709321041217,
%T A231273 26315271553053477373,261082718496449122051,2530297234481911294093,
%U A231273 5609403368997817686249127547,61628132164268458257532691681,354198989901889536240773677094747
%N A231273 Numerator of zeta(4n)/(zeta(2n) * Pi^(2n)).
%C A231273 Integer component of the numerator of a close variant of Euler's infinite prime product zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)-1), namely with all minus signs changed into plus signs, as follows: zeta(4n)/zeta(2n) = Product_{k>=1} (prime(k)^(2n))/(prime(k)^(2n)+1). The transcendental component is Pi^(2n).
%C A231273 For a detailed account of the results, including proof and relation to the zeta function, see Links for the PDF file submitted as supporting material.
%C A231273 The reference to Apostol is to a discussion of the equivalence of 1) zeta(2s)/zeta(s) and 2) a related infinite prime product, that is, Product_{sigma>1} prime(n)^s/(prime(n)^s + 1), with s being a complex variable such that s = sigma + i*t where sigma and t are real (following Riemann), using a type of proof different from the one posted below involving zeta(4n)/zeta(2n). On this, see also Hardy and Wright cited below. - _Leo Depuydt_, Nov 22 2013, Nov 27 2013
%C A231273 The background of the sequence is now described in the link below to L. Depuydt, The Prime Sequence ... . - _Leo Depuydt_, Aug 22 2014
%C A231273 From _Robert Israel_, Aug 22 2014: (Start)
%C A231273 Numerator of (-1)^n*B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!), where B(n) are the Bernoulli numbers (see A027641 and A027642).
%C A231273 Not the same as abs(A001067(2*n)): they differ first at n=17.
%C A231273 (End)
%D A231273 T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.
%D A231273 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fourth Edition, Clarendon Press, 1960, p. 255.
%H A231273 Leo Depuydt, <a href="/A231273/a231273_6.pdf">Details, including proof and relation to zeta function</a>
%p A231273 seq(numer((-1)^n*bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # _Robert Israel_, Aug 22 2014
%t A231273 Numerator[Table[Zeta[4n]/(Zeta[2n] * Pi^(2n)), {n, 0, 15}]] (* _T. D. Noe_, Nov 18 2013 *)
%Y A231273 Cf. A231327 (corresponding denominator).
%Y A231273 Cf. A114362 and A114363 (closely related results).
%Y A231273 Cf. A001067, A046968, A046988, A098087, A141590, A156036 (same number sequence, though in various transformations (alternation of signs, intervening numbers, and so on)).
%Y A231273 Cf. A027641 and A027642
%K A231273 nonn,frac
%O A231273 0,4
%A A231273 _Leo Depuydt_, Nov 07 2013