

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
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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 kalmost 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)^21)/(prime(n)^2+1) * Prod[from n = 1 to infinity] (semiprime(n)^2  1)/(semiprime(n)^2 + 1) * Prod[from n = 1 to infinity] (3almostprime(n)^2  1)/ (3almostprime(n)^2 + 1) * ... * Prod[from n = 1 to infinity] (kalmostprime(n)^2  1)/ (kalmostprime(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 (3almost 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. 47, 2004.


LINKS

Table of n, a(n) for n=0..66.
R. J. Mathar, HardyLittlewood 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*(2l1))/(2l1), where P_k(s) are the kalmost 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*=12/((p*q)^2+1))); x \\ Charles R Greathouse IV, Aug 15 2011


CROSSREFS

Cf. A001358, A090986, A114529A114536.
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



