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%I #20 Jan 15 2021 09:23:56
%S 7,5,4,4,9,9,7,0,1,7,0,9,5,1,4,0,7,8,3,5,5,7,1,8,1,6,8,9,5,0,5,4,1,9,
%T 8,7,0,2,5,0,7,7,6,4,4,3,5,8,7,2,2,3,3,8,9,0,9,9,7,9,9,1,6,4,2,8,4
%N Decimal expansion of a semiprime analog of a Ramanujan formula.
%C This is related to Ramanujan's surprising formula: Prod[from n = 1 to infinity] (prime(n)^2 - 1)/ (prime(n)^2 + 1) = 2/5 and we use it in finding A112407 as the semiprime analog. We also use: A090986 = Decimal expansion of Pi csch Pi = Prod[from n = 2 to infinity] (n^2 - 1)/(n^2 + 1).
%C Since every integer above 1 is a k-almost prime for some k, we factor the (n^2 - 1)/(n^2 + 1) infinite product and use Ramanujan's formula, to have: Prod[from n = 1 to infinity] (prime(n)^2-1)/(prime(n)^2+1) * Prod[from n = 1 to infinity] (semiprime(n)^2 - 1)/(semiprime(n)^2 + 1) * Prod[from n = 1 to infinity] (3-almostprime(n)^2 - 1)/ (3-almostprime(n)^2 + 1) * ... * Prod[from n = 1 to infinity] (k-almostprime(n)^2 - 1)/ (k-almostprime(n)^2 + 1) * ... = Prod[from n = 2 to infinity] (n^2 - 1)/(n^2 + 1) = pi csch pi as each integer appear once and only once in numerator and once and only once in denominator.
%C 2/5 is the first (Ramanujan, prime) term in this infinite product of infinite products. This here is the second (semiprime) term. A155799 is the third (3-almost prime) term. All of these have slow convergence.
%D Borwein, J.; Bailey, D.; and Girgensohn, R. "Two Products." Section 1.2 in Experimentation in Mathematics: Computational Paths to Discovery. Natick, MA: A. K. Peters, pp. 4-7, 2004.
%H R. J. Mathar, <a href="https://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, arXiv:0903.2514, Table 1, k=s=2.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HyperbolicCosecant.html">Hyperbolic Cosecant</a>
%F Decimal expansion of a = prod[from n = 1 to infinity] (semiprime(n)^2 - 1)/(semiprime(n)^2 + 1) = prod[from n = 1 to infinity] (A001358(n)^2 - 1)/(A001358(n)^2 + 1).
%F log a = -2*sum_{l=1..infinity} P_2(2*(2l-1))/(2l-1), where P_k(s) are the k-almost prime zeta functions of arXiv:0803.0900 [math.NT]. - _R. J. Mathar_, Jan 27 2009
%e 0.75449970170951407835571816895054...
%t _Robert G. Wilson v_: spQ[n_] := Plus @@ Last /@ FactorInteger@n == 2; p = 1; Do[If[spQ[n], p = N[p*(n^2 - 1)/(n^2 + 1), 64]], {n, 10}]; p
%o (PARI) A(lim)=my(x=1.);forprime(p=2,lim\2,forprime(q=2,min(p,lim\p),x*=1-2/((p*q)^2+1)));x \\ _Charles R Greathouse IV_, Aug 15 2011
%Y Cf. A001358, A090986, A114529-A114536.
%K cons,nonn
%O 0,1
%A _Jonathan Vos Post_, Dec 21 2005
%E Edited and extended by _R. J. Mathar_, Jan 27 2009
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