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Search: id:A112407
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| A112407 |
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Decimal expansion of a semiprime analogue of a Ramanujan formula. |
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+0
3
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| 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; cons; graph; listen)
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OFFSET
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0,1
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COMMENT
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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.
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REFERENCES
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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.
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LINKS
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Eric Weisstein's World of Mathematics, Infinite Product.
Eric Weisstein's World of Mathematics, Hyperbolic Cosecant
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FORMULA
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a = decimal expansion of Prod[from n = 1 to infinity] (semiprime(n)^2 - 1)/(semiprime(n)^2 + 1) = decimal expansion of 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
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EXAMPLE
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0.75449970170951407835571816895054...
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MATHEMATICA
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Robert G. Wilson v (rgwv(AT)rgwv.com): 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
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CROSSREFS
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Cf. A001358, A090986, A114529-A114536.
Sequence in context: A021061 A066960 A061827 this_sequence A154195 A019858 A109134
Adjacent sequences: A112404 A112405 A112406 this_sequence A112408 A112409 A112410
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KEYWORD
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cons,nonn
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AUTHOR
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Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 21 2005
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EXTENSIONS
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Edited and extended by R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jan 27 2009
Updated A-numbers in comments R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 12 2009
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