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 A231327 #129 Nov 04 2024 02:30:16
%S A231327 1,15,105,675675,34459425,16368226875,218517792968475,
%T A231327 30951416768146875,694097901592400930625,
%U A231327 23383376494609715287281703125,2289686345687357378035370971875,219012470258383844016431785453125,4791965046290912124048163518904807546875
%N A231327 Denominator of rational component of zeta(4n)/zeta(2n).
%C A231327 Denominator 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).
%C A231327 For a detailed account of the results in question, including proof and relation to the zeta function, see the PDF file submitted as supporting material in A231273.
%C A231327 The reference to Apostol below is 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). - _Leo Depuydt_, Nov 22 2013
%C A231327 Denominator of B(4*n)*4^n*(2*n)!/(B(2*n)*(4*n)!) where B(n) are the Bernoulli numbers (see A027641 and A027642). - _Robert Israel_, Aug 22 2014
%D A231327 T. M. Apostol, Introduction to Analytic Number Theory, Springer, 1976, p. 231.
%p A231327 seq(denom(bernoulli(4*n)*4^n*(2*n)!/(bernoulli(2*n)*(4*n)!)),n=0..100); # _Robert Israel_, Aug 22 2014
%t A231327 Denominator[Table[Zeta[4 n]/Zeta[2 n], {n, 0, 15}]] (* _T. D. Noe_, Nov 15 2013 *)
%Y A231327 Cf. A231273 (the corresponding numerator).
%Y A231327 Cf. A114362 and A114363 (closely related results).
%Y A231327 Cf. A001067, A046968, A046988, A098087, A141590, and A156036 (same number sequence as found in numerator, though in various transformations (alternation of sign, intervening numbers, and so on)).
%Y A231327 Cf. A027641 and A027642.
%K A231327 nonn,frac
%O A231327 0,2
%A A231327 _Leo Depuydt_, Nov 07 2013