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A065474 Decimal expansion of Product_{p prime} (1 - 2/p^2). 42
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET
0,1
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
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. [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).
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.
EXAMPLE
0.322634098939244670579531692548...
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 *)
PROG
(PARI) prodeulerrat(1 - 2/p^2) \\ Amiram Eldar, Mar 16 2021
CROSSREFS
Sequence in context: A106335 A218698 A352836 * A272332 A197586 A111702
KEYWORD
cons,nonn
AUTHOR
N. J. A. Sloane, Nov 19 2001
EXTENSIONS
Edited by Dean Hickerson, Sep 10 2002
More digits from Vaclav Kotesovec, Dec 18 2019
STATUS
approved

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Last modified March 29 11:14 EDT 2024. Contains 371278 sequences. (Running on oeis4.)