A369095 Decimal expansion of the asymptotic probability that 3 random integer 3 X 3 matrices generate the ring M_3(Z).

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

Offset

0,1

Links

Table of n, a(n) for n=0..104.

Steven Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 2020-2022. See p. 40.

Rostyslav V. Kravchenko, Marcin Mazur, and Bogdan V. Petrenko, On the smallest number of generators and the probability of generating an algebra, Algebra & Number Theory, Vol. 6, No. 2 (2012), pp. 243-291; arXiv preprint, arXiv:1001.2873 [math.RA], 2010.

Formula

Equals (1/(zeta(2)*zeta(3)*zeta(4))) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 1/p^5).

Example

0.79549366004648631320894021318758001965919253775693...

Mathematica

$MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{0, -1, -1, 0, 1}, {0, 2, 3, -2, -10}, m]; RealDigits[Exp[NSum[Indexed[c, n]*(PrimeZetaP[n])/n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]]/(Zeta[2]*Zeta[3]*Zeta[4]), 10, 120][[1]]

Prog

(PARI) prodeulerrat(1 + 1/p^2 + 1/p^3 - 1/p^5)/(zeta(2)*zeta(3)*zeta(4))

Crossrefs

Cf. A002117, A013661, A013662, A369094.

Sequence in context: A216104 A093206 A070693 * A198417 A363687 A215736

Adjacent sequences: A369092 A369093 A369094 * A369096 A369097 A369098

Keyword

nonn,cons

Author

Amiram Eldar, Jan 13 2024

Status

approved