A065474 Decimal expansion of Product_{p prime} (1 - 2/p^2).

3, 2, 2, 6, 3, 4, 0, 9, 8, 9, 3, 9, 2, 4, 4, 6, 7, 0, 5, 7, 9, 5, 3, 1, 6, 9, 2, 5, 4, 8, 2, 3, 7, 0, 6, 6, 5, 7, 0, 9, 5, 0, 5, 7, 9, 6, 6, 5, 8, 3, 2, 7, 0, 9, 9, 6, 1, 8, 1, 1, 2, 5, 2, 4, 5, 3, 2, 5, 0, 0, 6, 3, 4, 8, 6, 2, 4, 4, 6, 0, 9, 8, 8, 4, 5, 2, 3, 4, 8, 1, 5, 6, 8, 5, 6, 3, 7, 5, 5, 2, 1, 7, 7, 2, 7, 3

Offset: 0

Author

N. J. A. Sloane, Nov 19 2001

Keywords

• cons
• nonn

Comments

Density of A007674, squarefree n such that n + 1 is squarefree. - Charles R Greathouse IV, Aug 10 2011

Product_{k>=1} (1 - 2/k^2) = sin(sqrt(2)*Pi) / (sqrt(2)*Pi). - Vaclav Kotesovec, May 23 2020

The asymptotic probability that, for two integers k and m, 0 < k <= m, we have gcd(k*(k+1), m) = 1 (when k and m are chosen at random in the range 1..n and n->oo) (Tóth and Sándor, 1989). - Amiram Eldar, Apr 29 2023

Examples

			0.322634098939244670579531692548...
		

References

• Henri Cohen, Number Theory, Volume II: Analytic and Modern Tools, GTM Vol. 240, Springer, 2007; see pp. 208-209.

Links

• 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. 163 and 184.
• R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], 2009-2011, constant F_1^(2).
• G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
• L. Tóth and J. Sándor, An asymptotic formula concerning a generalized Euler function, Fibonacci Quarterly, Vol. 27, No. 2 (1989), pp. 176-180.
• Eric Weisstein's World of Mathematics, Squarefree.
• Eric Weisstein's World of Mathematics, Prime Products.
• Zack Wolske, Number Fields with Large Minimal Index, Ph.D. Thesis, University of Toronto, 2018.

Crossrefs

Cf. A007674, A058026, A065493, A074893.

Programs

Mathematica

$MaxExtraPrecision = 800; digits = 98; terms = 800; P[n_] := PrimeZetaP[n]; LR = Join[{0, 0}, LinearRecurrence[{0, 2}, {-4, 0}, terms + 10]]; r[n_Integer] := LR[[n]]; Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 18 2016 *)

PARI

prodeulerrat(1 - 2/p^2) \\ Amiram Eldar, Mar 16 2021

Extensions

Edited by Dean Hickerson, Sep 10 2002

More digits from Vaclav Kotesovec, Dec 18 2019