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 A114362 #45 Mar 04 2023 05:06:38
%S A114362 2,2,6,691,7234,523833,3545461365,3392780147,15418642082434,
%T A114362 26315271553053477373,261082718496449122051,2530297234481911294093,
%U A114362 39265823582984723803743892829,61628132164268458257532691681
%N A114362 Numerator of zeta(4n)/zeta(2n)^2 (with a(0)=2 instead of -2).
%C A114362 zeta(4n)/zeta(2n)^2 is a rational value expressible in term of Bernoulli's numbers (A027641).
%C A114362 Conjecture: if an integer n > 1 is odd, then zeta(2n)/zeta(n)^2 is irrational. Cf. W. Kohnen (link) and my conjecture in A348829. - _Thomas Ordowski_, Jan 05 2022
%C A114362 Conjecture: (1 - t(n))/(1 + t(n)) = 1/2^n + 1/3^n + 1/5^n + 1/7^n + O(1/11^n), where t(n) = zeta(2n)/zeta(n)^2. Cf. A348829. - _Thomas Ordowski_, Nov 13 2022
%H A114362 Seiichi Manyama, <a href="/A114362/b114362.txt">Table of n, a(n) for n = 0..158</a>
%H A114362 Winfried Kohnen, <a href="https://doi.org/10.1007/BF02864395">Transcendence conjectures about periods of modular forms and rational structures on spaces of modular forms</a>, Proceedings of the Indian Academy of Sciences-Mathematical Sciences, Vol. 99, No. 3 (1989), pp. 231-233.
%H A114362 Herbert Wilf, <a href="http://www.jstor.org/stable/4145136">Problem 11068</a>, The American Mathematical Monthly, Vol. 111, No. 3 (2004), p. 259; <a href="http://www.jstor.org/stable/30037619">Think Rationally</a>, Solution to Problem 11068 by Kenneth E. Schilling, ibid., Vol. 112, No. 9 (2005), pp. 844-845.
%F A114362 Product_{p primes} (p^{2n}-1)/(p^{2n}+1) = zeta(4n)/zeta(2n)^2.
%F A114362 For n > 0, a(n) = Numerator((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_, Jan 05 2022
%F A114362 From _Amiram Eldar_, Mar 04 2023: (Start)
%F A114362 a(n)/A114363(n) = -2*B(4*n)/(binomial(4*n,*2n)*B(2*n)) = -2*(A027641(4*n)/A027642(4*n))/(A000984(2*n)*A027641(2*n)/A027642(2*n)), for n >= 1, where B(n) is the n-th Bernoulli number.
%F A114362 A114363(n)/a(n) = Sum_{x in Q+} 1/f(x)^(2*n), for n >= 1, where Q+ is the set of the positive rational numbers, and if x = k/m in lowest terms, then f(x) = k*m (Wilf, 2004). (End)
%e A114362 2/1, 2/5, 6/7, 691/715, 7234/7293, 523833/524875, 3545461365/3547206349, ...
%t A114362 a[n_] := Numerator[Zeta[4*n]/Zeta[2*n]^2]; a[0] = 2; Array[a, 14, 0] (* _Amiram Eldar_, Mar 04 2023 *)
%o A114362 (PARI) z(n)=bernfrac(2*n)*(-1)^(n - 1)*2^(2*n-1)/(2*n)!;
%o A114362 a(n)=if(n<1,2,numerator(z(2*n)/z(n)^2))
%Y A114362 Cf. A000984, A027641, A027642, A114363 (denominators), A348829, A348830.
%K A114362 frac,nonn
%O A114362 0,1
%A A114362 _Benoit Cloitre_, Feb 09 2006; corrected Feb 22 2006