A369095 - OEIS

%I #7 Jan 13 2024 10:44:53

%S 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,

%T 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,

%U 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

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

%H Steven Finch, <a href="https://arxiv.org/abs/2001.00578">Errata and Addenda to Mathematical Constants</a>, arXiv:2001.00578 [math.HO], 2020-2022. See p. 40.

%H Rostyslav V. Kravchenko, Marcin Mazur, and Bogdan V. Petrenko, <a href="http://dx.doi.org/10.2140/ant.2012.6.243">On the smallest number of generators and the probability of generating an algebra</a>, Algebra & Number Theory, Vol. 6, No. 2 (2012), pp. 243-291; <a href="https://arxiv.org/abs/1001.2873">arXiv preprint</a>, arXiv:1001.2873 [math.RA], 2010.

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

%e 0.79549366004648631320894021318758001965919253775693...

%t $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]]

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

%Y Cf. A002117, A013661, A013662, A369094.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Jan 13 2024