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

0,1

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

References

Gerald Tenenbaum, Introduction to Analytic and Probabilistic Number Theory, 3rd edition, American Mathematical Society, 2015, page 59, exercise 55 and 56.

Links

Table of n, a(n) for n=0..105.

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.

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

Example

0.2867474284344787341078927127898384...

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 *)

Prog

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

Crossrefs

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

Sequence in context: A199504 A368645 A019914 * A054029 A197589 A124356

Adjacent sequences: A065470 A065471 A065472 * A065474 A065475 A065476

Keyword

cons,nonn

Author

N. J. A. Sloane, Nov 19 2001

Extensions

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

More digits from Vaclav Kotesovec, Dec 19 2019

Status

approved