A320027 Decimal expansion of the probability that an integer 4-tuple is pairwise unitary coprime.

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

Offset

0,2

Comments

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

References

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

Links

Table of n, a(n) for n=0..86.

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).

Formula

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).

Example

0.137310651809073591871587470612435012319854472214516...

Mathematica

$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]]

Prog

(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

Crossrefs

Cf. A077610, A306071, A319592.

Sequence in context: A248214 A144713 A233380 * A134731 A133368 A153027

Adjacent sequences: A320024 A320025 A320026 * A320028 A320029 A320030

Keyword

nonn,cons

Author

Amiram Eldar, Aug 27 2019

Status

approved