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 A112407 Decimal expansion of a semiprime analog of a Ramanujan formula. 3
 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, 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 (list; constant; graph; refs; listen; history; text; internal format)
 OFFSET 0,1 COMMENTS 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). 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. 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. REFERENCES 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. LINKS R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514, Table 1, k=s=2. Eric Weisstein's World of Mathematics, Infinite Product. Eric Weisstein's World of Mathematics, Hyperbolic Cosecant FORMULA 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). 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 EXAMPLE 0.75449970170951407835571816895054... MATHEMATICA 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 PROG (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 CROSSREFS Cf. A001358, A090986, A114529-A114536. Sequence in context: A066960 A061827 A273841 * A154195 A280870 A019858 Adjacent sequences:  A112404 A112405 A112406 * A112408 A112409 A112410 KEYWORD cons,nonn AUTHOR Jonathan Vos Post, Dec 21 2005 EXTENSIONS Edited and extended by R. J. Mathar, Jan 27 2009 STATUS approved

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Last modified May 11 03:09 EDT 2021. Contains 343784 sequences. (Running on oeis4.)