A319593 Decimal expansion of the probability that an integer triple is pairwise unitary coprime.

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

Offset: 0

Author

Amiram Eldar, Aug 27 2019

Keywords

• nonn
• cons

Comments

Two numbers are unitary coprime if their largest common unitary divisor is 1.

Examples

			0.552306904157942811183227347309264708535455831404497...
		

References

• Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, p. 54.

Links

• 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).

Crossrefs

Cf. A065473, A077610, A306071.

Programs

Mathematica

$MaxExtraPrecision = 1000; nm = 1000; f[x_] := 1 - 4*x^2 + 7*x^3 - 9*x^4 + 8*x^5 - 2*x^6 - 3*x^7 + 2*x^8; c = LinearRecurrence[{-1, 3, -4, 5, -3, -1, 2}, {0, -8, 21, -68, 180, -503, 1428}, nm]; RealDigits[f[1/2] * f[1/3] * Zeta[2] * Zeta[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]]

PARI

zeta(2) * zeta(3) * prodeulerrat(1-4/p^2+7/p^3-9/p^4+8/p^5-2/p^6-3/p^7+2/p^8) \\ Amiram Eldar, Jun 29 2023

Formula

Equals zeta(2) * zeta(3) * Product_{p prime} (1 - 4/p^2 + 7/p^3 - 9/p^4 + 8/p^5 - 2/p^6 - 3/p^7 + 2/p^8).