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A065474
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Decimal expansion of Product_{p prime} (1 - 2/p^2).
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42
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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
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OFFSET
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0,1
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COMMENTS
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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
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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.
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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. 163 and 184.
Eric Weisstein's World of Mathematics, Squarefree.
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EXAMPLE
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0.322634098939244670579531692548...
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MATHEMATICA
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$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 *)
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PROG
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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