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%I #63 Jun 13 2021 13:31:08
%S 2,8,6,7,4,7,4,2,8,4,3,4,4,7,8,7,3,4,1,0,7,8,9,2,7,1,2,7,8,9,8,3,8,4,
%T 4,6,4,3,4,3,3,1,8,4,4,0,9,7,0,5,6,9,9,5,6,4,1,4,7,7,8,5,9,3,3,6,6,5,
%U 2,2,4,3,1,3,1,9,4,3,2,5,8,2,4,8,9,1,2,6,8,2,5,5,3,7,4,2,3,7,4,6,8,5,3,6,4,7
%N Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).
%C Also decimal expansion of the probability that an integer triple (x, y, z) is pairwise coprime. - _Charles R Greathouse IV_, Nov 14 2011
%C The probability that 2 numbers chosen at random are coprime, and both squarefree (Delange, 1969). - _Amiram Eldar_, Aug 04 2020
%D Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edition, American Mathematical Society, 2015, page 59, exercise 55 and 56.
%H Juan Arias de Reyna, R. Heyman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL18/Heyman/heyman6.html">Counting Tuples Restricted by Pairwise Coprimality Conditions</a>, J. Int. Seq. 18 (2015) 15.10.4
%H Tim Browning, <a href="https://jtnb.centre-mersenne.org/article/JTNB_2011__23_3_579_0.pdf">The divisor problem for binary cubic forms</a>, Journal de théorie des nombres de Bordeaux, Vol. 23, No. 3 (2011), pp. 579-602; <a href="http://arxiv.org/abs/1006.3476">arXiv preprint</a>, arXiv:1006.3476 [math.NT], 2010.
%H Hubert Delange, <a href="https://doi.org/10.1016/0022-314X(69)90045-6">On some sets of pairs of positive integers</a>, Journal of Number Theory, Vol. 1, No. 3 (1969), pp. 261-279. See p. 277.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 181.
%H G. Niklasch, <a href="http://guests.mpim-bonn.mpg.de/moree/Moree.en.html">Some number theoretical constants: 1000-digit values</a>.
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [cached copy]
%H László Tóth, <a href="http://www.fq.math.ca/Scanned/40-1/toth.pdf">The probability that k positive integers are pairwise relatively prime</a>, Fibonacci Quart., Vol. 40 (2002), pp. 13-18.
%H László Tóth, <a href="https://arxiv.org/abs/2006.12438">Another generalization of Euler's arithmetic function and of Menon's identity</a>, arXiv:2006.12438 [math.NT], 2020. See p. 3.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CarefreeCouple.html">Carefree Couple</a>.
%F Equals Prod_{p prime} (1 - 1/p)^2*(1 + 2/p). - _Michel Marcus_, Apr 16 2016
%F The constant c in Sum_{k<=x} mu(k)^2 * 2^omega(k) = c * x * log(x) + O(x), where mu is A008683 and omega is A001221, and in Sum_{k<=x} 3^omega(k) = (1/2) * c * x * log(x)^2 + O(x*log(x)) (see Tenenbaum, 2015). - _Amiram Eldar_, May 24 2020
%F Equals A065472 * A227929 = A065472 / A098198. - _Amiram Eldar_, Aug 04 2020
%e 0.2867474284344787341078927127898384...
%t digits = 100; NSum[-(2+(-2)^n)*PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2 digits, WorkingPrecision -> 2 digits, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Apr 11 2016 *)
%o (PARI) prodeulerrat(1 - (3*p-2)/(p^3)) \\ _Amiram Eldar_, Mar 17 2021
%Y Cf. A065472, A069201, A069212, A074816, A074823, A078073, A098198, A227929, A299822.
%K cons,nonn
%O 0,1
%A _N. J. A. Sloane_, Nov 19 2001
%E Name corrected by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 03 2003
%E More digits from _Vaclav Kotesovec_, Dec 19 2019
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