OFFSET
0,2
REFERENCES
Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.5.1 Carefree Couples, p. 110.
LINKS
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.
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
KEYWORD
nonn,cons
AUTHOR
Jean-François Alcover, May 15 2016
STATUS
approved