A065473 Decimal expansion of the strongly carefree constant: Product_{p prime} (1 - (3*p-2)/(p^3)).

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, 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, 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

Offset: 0

Author

N. J. A. Sloane, Nov 19 2001

Keywords

• cons
• nonn

Comments

Also decimal expansion of the probability that an integer triple (x, y, z) is pairwise coprime. - Charles R Greathouse IV, Nov 14 2011

The probability that 2 numbers chosen at random are coprime, and both squarefree (Delange, 1969). - Amiram Eldar, Aug 04 2020

Examples

			0.2867474284344787341078927127898384...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 1.6, p. 41.
• Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edition, American Mathematical Society, 2015, page 59, exercise 55 and 56.

Links

• Juan Arias de Reyna, R. Heyman, Counting Tuples Restricted by Pairwise Coprimality Conditions, J. Int. Seq. 18 (2015) 15.10.4
• Tim Browning, The divisor problem for binary cubic forms, Journal de théorie des nombres de Bordeaux, Vol. 23, No. 3 (2011), pp. 579-602; arXiv preprint, arXiv:1006.3476 [math.NT], 2010.
• Hubert Delange, On some sets of pairs of positive integers, Journal of Number Theory, Vol. 1, No. 3 (1969), pp. 261-279. See p. 277.
• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 181.
• G. Niklasch, Some number theoretical constants: 1000-digit values.
• G. Niklasch, Some number theoretical constants: 1000-digit values. [cached copy]
• László Tóth, The probability that k positive integers are pairwise relatively prime, Fibonacci Quart., Vol. 40 (2002), pp. 13-18.
• László Tóth, Another generalization of Euler's arithmetic function and of Menon's identity, arXiv:2006.12438 [math.NT], 2020. See p. 3.
• Eric Weisstein's World of Mathematics, Carefree Couple.

Crossrefs

Cf. A065472, A069201, A069212, A074816, A074823, A078073, A098198, A227929, A299822.

Programs

Mathematica

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 *)

PARI

prodeulerrat(1 - (3*p-2)/(p^3)) \\ Amiram Eldar, Mar 17 2021

Formula

Equals Prod_{p prime} (1 - 1/p)^2*(1 + 2/p). - Michel Marcus, Apr 16 2016

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

Equals A065472 * A227929 = A065472 / A098198. - Amiram Eldar, Aug 04 2020

Extensions

Name corrected by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 03 2003

More digits from Vaclav Kotesovec, Dec 19 2019