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A005596
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Decimal expansion of Artin's constant Product_{p=prime} (1-1/(p^2-p)).
(Formerly M2608)
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84
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3, 7, 3, 9, 5, 5, 8, 1, 3, 6, 1, 9, 2, 0, 2, 2, 8, 8, 0, 5, 4, 7, 2, 8, 0, 5, 4, 3, 4, 6, 4, 1, 6, 4, 1, 5, 1, 1, 1, 6, 2, 9, 2, 4, 8, 6, 0, 6, 1, 5, 0, 0, 4, 2, 0, 9, 4, 7, 4, 2, 8, 0, 2, 4, 1, 7, 3, 5, 0, 1, 8, 2, 0, 4, 0, 0, 2, 8, 0, 8, 2, 3, 4, 4, 3, 0, 4, 3, 1, 7, 0, 8, 7, 2, 5, 0, 5, 6, 8, 9, 8, 1, 6, 0, 3
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OFFSET
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0,1
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COMMENTS
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On Simon Plouffe's web page (and in the book freely available at Gutenberg project) the value is given with an error of +1e-31, as "...651641..." instead of "...641641...". In the reference [Wrench, 1961] cited there, these digits are correct. They are also correct on the Plouffe's Inverter page, as computed by Oliveira e Silva, who comments it took 1 hour at 200 MHz with Mathematica. Using Amiram Eldar's PARI program, the same 500 digits are computed instantly (less than 0.1 sec). - M. F. Hasler, Apr 20 2021
Named after the Austrian mathematician Emil Artin (1898-1962). - Amiram Eldar, Jun 20 2021
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REFERENCES
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Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C7).
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FORMULA
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Equals Sum_{k>=1} mu(k)/(k*phi(k)), where mu is the Moebius function (A008683) and phi is the Euler totient function (A000010). - Amiram Eldar, Mar 11 2020
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EXAMPLE
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0.37395581361920228805472805434641641511162924860615...
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MATHEMATICA
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a = Exp[-NSum[ (LucasL[n] - 1)/n PrimeZetaP[n], {n, 2, Infinity}, PrecisionGoal -> 500, WorkingPrecision -> 500, NSumTerms -> 100000]]; RealDigits[a, 10, 111][[1]] (* Robert G. Wilson v, Sep 03 2014 taken from Mathematica's Help file on PrimeZetaP *)
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PROG
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(PARI) prodinf(n=2, 1/zeta(n)^(sumdiv(n, d, moebius(n/d)*(fibonacci(d-1)+fibonacci(d+1)))/n)) \\ Charles R Greathouse IV, Aug 27 2014
(PARI) prodeulerrat(1-1/(p^2-p)) \\ Amiram Eldar, Mar 12 2021
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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More terms from Tomás Oliveira e Silva (http://www.ieeta.pt/~tos)
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STATUS
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approved
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