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A167864
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Decimal expansion of Selberg-Delange constant Product_{prime p > 2} (1 + 1/(p(p-2)))
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2
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1, 5, 1, 4, 7, 8, 0, 1, 2, 8, 1, 3, 7, 4, 9, 1, 2, 5, 7, 7, 9, 0, 9, 1, 9, 2, 5, 5, 6, 4, 9, 4, 7, 4, 8, 9, 2, 4, 1, 5, 2, 7, 0, 1, 5, 8, 2, 8, 6, 2, 1, 4, 3, 9, 5, 3, 5, 7, 4, 8, 4, 2, 7, 1, 4, 8, 4, 9, 3, 2, 2, 0, 9, 8, 1, 5, 6, 1, 1, 5, 8, 1, 0, 8, 7, 7, 5, 8, 5, 3, 8, 2, 7, 6, 9, 8, 0, 7, 6, 7, 7, 6, 5, 6, 2
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OFFSET
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1,2
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COMMENTS
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Coefficient in formulas for the distribution of integers with a fixed number of prime factors.
Reciprocal of the twin prime constant A005597. See A005597 for links and additional references and comments.
Numerators of partial products are A062271. Denominators are A062270. (End)
An analog for primes of Wallis' product pi/2 = Product_{n=1 to oo} (2n)^2/(2n-1)(2n+1), because A167864 = Product_{prime p>2} (p-1)^2/(p-2)p.
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REFERENCES
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M. Balazard, H. Delange and J.-L. Nicolas, Sur le nombre de facteurs premiers des entiers, C. R. Acad. Sci., Paris, Ser. I 306 (1988) 511-514. [From Jonathan Sondow, Nov 17 2009]
H. Delange, Sur des formules de Atle Selberg, Acta Arith. 19 (1971) 105-146.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 84-93.
J.-L. Nicolas, Sur la distribution des nombres entiers ayant une quantite fixee de facteurs premiers, Acta Arith. 44 (1984) 191-200.
A. Selberg, Note on a paper by L. G. Sathe, J. Indian Math. Soc. 18 (1954) 83-87.
G. Tenenbaum, Introduction to analytic and probabilistic number theory, Cambridge University Press, 1995, p. 206.
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LINKS
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Table of n, a(n) for n=1..105.
S. R. Finch, Mathematical Constants, Errata and AddendaSec. 2.1.
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FORMULA
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A167864 = 1/A005597.
Equals Product_{prime p>2} (p-1)^2/(p-2)p = (2^2/1*3)(4^2/3*5)(6^2/5*7)(10^2/9*11)....
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EXAMPLE
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Product_{prime p > 2} (1 + 1/(p(p-2))) = 1.51477...
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MATHEMATICA
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s[n_] := (1/n)* N[Sum[MoebiusMu[d]*2^(n/d), {d, Divisors[n]}], 160]; C2 = (175/256)*Product[(Zeta[ n]*(1 - 2^(-n))*(1 - 3^(-n))*(1 - 5^(-n))*(1 - 7^(-n)))^(-s[ n]), {n, 2, 160}]; RealDigits[1/C2][[1]][[1 ;; 105]] (* Jean-François Alcover, Oct 30 2012, after Pari program in A005597 *)
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CROSSREFS
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Cf. A005597.
Sequence in context: A216157 A216851 A179290 * A011301 A180132 A105664
Adjacent sequences: A167861 A167862 A167863 * A167865 A167866 A167867
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KEYWORD
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cons,nonn
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AUTHOR
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Jonathan Sondow, Nov 13 2009, Nov 17 2009
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STATUS
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approved
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