A206256 Decimal expansion of Product_{p prime} (1 - 3/p^2).

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

Offset: 0

Author

N. J. A. Sloane, Feb 05 2012, based on a posting by Warren Smith to the Math Fun Mailing List, Feb 04 2012

Keywords

• nonn
• cons

Comments

For a randomly selected number k, this is the probability that k, k+1, k+2 all are squarefree.

Examples

			0.1254869809058...
		

Links

• Leon Mirsky, Note on an asymptotic formula connected with r-free integers, The Quarterly Journal of Mathematics, Vol. os-18, No. 1 (1947), pp. 178-182.
• Leon Mirsky, Arithmetical pattern problems relating to divisibility by rth powers, Proceedings of the London Mathematical Society, Vol. s2-50, No. 1 (1949), pp. 497-508.

Crossrefs

Cf. A059956, A065474, A007675, A335131, A370600.

Programs

Maple

# See A175640 using efact := 1-3/p^2. - R. J. Mathar, Mar 22 2012

Mathematica

$MaxExtraPrecision = 500; m = 500; c = LinearRecurrence[{0, 3}, {0, -6}, m]; RealDigits[(1/4) * Exp[NSum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]] (* Amiram Eldar, Oct 01 2019 *)

PARI

prodeulerrat(1 - 3/p^2) \\ Amiram Eldar, Mar 16 2021

Extensions

More terms from Amiram Eldar, Oct 01 2019

More terms from Vaclav Kotesovec, Dec 17 2019