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A320027
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Decimal expansion of the probability that an integer 4-tuple is pairwise unitary coprime.
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0
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1, 3, 7, 3, 1, 0, 6, 5, 1, 8, 0, 9, 0, 7, 3, 5, 9, 1, 8, 7, 1, 5, 8, 7, 4, 7, 0, 6, 1, 2, 4, 3, 5, 0, 1, 2, 3, 1, 9, 8, 5, 4, 4, 7, 2, 2, 1, 4, 5, 1, 6, 1, 5, 4, 3, 9, 9, 3, 9, 4, 4, 4, 4, 1, 5, 0, 4, 5, 6, 8, 1, 9, 6, 2, 8, 9, 6, 0, 8, 2, 7, 5, 7, 5, 4, 5, 6
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OFFSET
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0,2
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COMMENTS
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Two numbers are unitary coprime if their largest common unitary divisor is 1.
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REFERENCES
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Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 54.
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LINKS
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László Tóth, Multiplicative arithmetic functions of several variables: a survey, in Themistocles M. Rassias and Panos M. Pardalos (eds.), Mathematics Without Boundaries, Springer, New York, NY, 2014, pp. 483-514 (see p. 509), preprint, arXiv:1310.7053 [math.NT], 2013-2014 (see p. 22).
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FORMULA
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Equals zeta(2)^2 * zeta(3) * zeta(4) * Product_{p prime} (1 - 8/p^2 + 3/p^3 + 27/p^4 - 24/p^5 - 14/p^6 - 3/p^7 + 37/p^8 - 30/p^9 + 42/p^10 - 33/p^11 - 41/p^12 + 78/p^13 - 44/p^14 + 9/p^15).
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EXAMPLE
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0.137310651809073591871587470612435012319854472214516...
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MATHEMATICA
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$MaxExtraPrecision = 1000; nm = 1000; f[x_] := 1 - 8*x^2 + 3*x^3 + 27*x^4 - 24*x^5 - 14*x^6 - 3*x^7 + 37*x^8 - 30*x^9 + 42*x^10 - 33*x^11 - 41*x^12 + 78*x^13 - 44*x^14 + 9*x^15; c = LinearRecurrence[{-3, 2, 11, -3, -16, -14, 6, 7, 19, 0, -17, 9}, {0, -16, 9, -20, 0, 161, -588, 2116, -5859, 15104, -34716, 70609}, nm]; RealDigits[Zeta[2]^2*Zeta[3]*Zeta[4]*f[1/2]*f[1/3]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k - 1/3^k)/k, {k, 2, nm}, NSumTerms -> nm, WorkingPrecision -> nm]], 10, 100][[1]]
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PROG
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(PARI) zeta(2)^2 * zeta(3) * zeta(4) * prodeulerrat(1-8/p^2+3/p^3+27/p^4-24/p^5-14/p^6-3/p^7+37/p^8-30/p^9+42/p^10-33/p^11-41/p^12+78/p^13-44/p^14+9/p^15) \\ Amiram Eldar, Jun 29 2023
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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