A065464 Decimal expansion of Product_{p prime} (1 - (2*p-1)/p^3).

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

Offset

0,1

Comments

Sum_{n <= x} A189021(n) ~ kx, where k is this constant. - Charles R Greathouse IV, Jan 24 2018

The probability that a number chosen at random is squarefree and coprime to another randomly chosen random (see Schroeder, 2009). - Amiram Eldar, May 23 2020, corrected Aug 04 2020

References

Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.5.1, p. 110.

Manfred Schroeder, Number Theory in Science and Communication, 5th edition, Springer, 2009, page 59.

Links

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

R. J. Mathar, Hardy-Littlewood Constants Embedded into Infinite Products over All Positive Integers, arXiv:0903.2514 [math.NT], 2009-2011; Equation (116).

G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]

Eric Weisstein's World of Mathematics, Carefree Couple.

Formula

Equals A065463 divided by A013661. - R. J. Mathar, Mar 22 2011

Equals A065473 divided by A065480. - R. J. Mathar, May 02 2019

Equals (6/Pi^2)^2 * Product_{p prime} (1 + 1/(p^3 + p^2 - p - 1)) = 1.1587609... * (6/Pi^2)^2. - Amiram Eldar, May 23 2020

Equals lim_{m->oo} (1/m) * Sum_{k==1..m} (phi(k)/k)^2, where phi is the Euler totient function (A000010). - Amiram Eldar, Mar 12 2021

Example

0.428249505677094440218765707581823546...

Mathematica

$MaxExtraPrecision = 800; digits = 98; terms = 2000; LR = Join[{0, 0}, LinearRecurrence[{-2, 0, 1}, {-2, 3, -6}, terms+10]]; r[n_Integer] := LR[[n]]; (6/Pi^2)*Exp[NSum[r[n]*(PrimeZetaP[n-1]/(n-1)), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10, Method -> "AlternatingSigns"]] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

Prog

(PARI) prodeulerrat(1 - (2*p-1)/p^3) \\ Amiram Eldar, Mar 12 2021

Crossrefs

Cf. A000010, A057434, A078078, A056671.

Cf. A065463, A013661.

Cf. A065473, A065480.

Sequence in context: A112152 A211883 A083489 * A201400 A040015 A144926

Adjacent sequences: A065461 A065462 A065463 * A065465 A065466 A065467

Keyword

cons,nonn

Author

N. J. A. Sloane, Nov 19 2001

Extensions

More digits from Vaclav Kotesovec, Dec 18 2019

Status

approved