A005596 Decimal expansion of Artin's constant Product_{p=prime} (1-1/(p^2-p)).

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

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• cons

Comments

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

Examples

			0.37395581361920228805472805434641641511162924860615...
		

References

• Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.
• John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See p. 169.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Harry J. Smith, Table of n, a(n) for n = 0..1000
• Ivan Cherednik, A note on Artin's constant, arXiv:0810.2325 [math.NT], 2008.
• Henri Cohen, High-precision computation of Hardy-Littlewood constants, (1998).
• Henri Cohen, High-precision computation of Hardy-Littlewood constants. [pdf copy, with permission]
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 156 (constant C7).
• R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2001; constant A_1^(1).
• Pieter Moree, Artin's primitive root conjecture - a survey, arXiv:math/0412262 [math.NT], 2004-2012.
• Pieter Moree, The formal series Witt transform, Discr. Math., Vol. 295, No. 1-3 (2005), pp. 143-160. See p. 159.
• G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
• G. Niklasch, Artin's constant.
• Simon Plouffe, The Artin's Constant=product(1-1/(p**2-p), p=prime) [backup on web.archive.org; chapter 8 of the free Gutenberg.org/ebooks/634]. [Warning: the value given in this reference is incorrect, cf. comment!]
• Tomás Oliveira e Silva and Plouffe's Inverter, The first 500 digits of Artin's constant.
• Eric Weisstein's World of Mathematics, Artin's constant.
• Eric Weisstein's World of Mathematics, Full Reptend Prime.
• R. G. Wilson v, Letter to N. J. A. Sloane, Aug. 1993.
• John W. Wrench, Jr., Evaluation of Artin's constant and the twin-prime constant, Math. Comp., Vol. 15, No. 76 (1961), pp. 396-398.
• Index entries for sequences related to Artin's conjecture.

Crossrefs

Cf. A048296, A065414, A001913, A001122.

Programs

Mathematica

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 *)

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

Formula

Equals Product_{j>=2} 1/Zeta(j)^A006206(j), where Zeta = A013661, A002117 etc. is Riemann's zeta function. - R. J. Mathar, Feb 14 2009

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

Equals 1/A065488. - Vaclav Kotesovec, Jul 17 2021

Extensions

More terms from Tomás Oliveira e Silva (http://www.ieeta.pt/~tos)