A273093 Decimal expansion of the probability that three positive integers are pairwise not coprime.

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

Offset

0,2

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.5.1 Carefree Couples, p. 110.

Links

Table of n, a(n) for n=0..99.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020, p. 15.

Randell Heyman, Pairwise non-coprimality of triples, arXiv preprint, arXiv:1309.5578 [math.NT], 2013-2014.

Randell Heyman, Topics in divisibility: pairwise coprimality, the GCD of shifted sets and polynomial irreducibility, PhD thesis, University of New South Wales, 2015.

Pieter Moree, Counting carefree couples, arXiv:math/0510003 [math.NT], 2005-2014.

Eric Weisstein's MathWorld, Carefree Couple.

Formula

Equals 1 - 18/Pi^2 + 3P - Q, where P is A065464 and Q is A065473.

Example

0.1742197830347247005585740721805346916511057518703135572332637051646...

Mathematica

$MaxExtraPrecision = 1000; digits = 100; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms + 10]]; r[n_Integer] := LR[[n]];
P = (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n - 1]/(n - 1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10, Method -> "AlternatingSigns"]];
Q = NSum[-(2 + (-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 3digits, Method -> "AlternatingSigns"]//Exp;
F3 = 1 - 18/Pi^2 + 3P - Q;
RealDigits[F3, 10, digits][[1]]

Prog

(PARI) 1 - 3/zeta(2) + 3 * prodeulerrat(1 - (2*p-1)/p^3) - prodeulerrat(1 - (3*p-2)/p^3) \\ Amiram Eldar, Mar 03 2021

Crossrefs

Cf. A065464, A065473, A229099.

Sequence in context: A216261 A188628 A021578 * A071185 A165244 A198356

Adjacent sequences: A273090 A273091 A273092 * A273094 A273095 A273096

Keyword

nonn,cons

Author

Jean-François Alcover, May 15 2016

Status

approved