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%I #40 Apr 16 2024 03:06:23
%S 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,
%T 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,
%U 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
%N Decimal expansion of Product_{p prime} (1 - 3/p^2).
%C For a randomly selected number k, this is the probability that k, k+1, k+2 all are squarefree.
%H Leon Mirsky, <a href="https://doi.org/10.1093/qmath/os-18.1.178">Note on an asymptotic formula connected with r-free integers</a>, The Quarterly Journal of Mathematics, Vol. os-18, No. 1 (1947), pp. 178-182.
%H Leon Mirsky, <a href="https://doi.org/10.1112/plms/s2-50.7.497">Arithmetical pattern problems relating to divisibility by rth powers</a>, Proceedings of the London Mathematical Society, Vol. s2-50, No. 1 (1949), pp. 497-508.
%e 0.1254869809058...
%p # See A175640 using efact := 1-3/p^2. - _R. J. Mathar_, Mar 22 2012
%t $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 *)
%o (PARI) prodeulerrat(1 - 3/p^2) \\ _Amiram Eldar_, Mar 16 2021
%Y Cf. A059956, A065474, A007675, A335131, A370600.
%K nonn,cons,changed
%O 0,2
%A _N. J. A. Sloane_, Feb 05 2012, based on a posting by Warren Smith to the Math Fun Mailing List, Feb 04 2012
%E More terms from _Amiram Eldar_, Oct 01 2019
%E More terms from _Vaclav Kotesovec_, Dec 17 2019
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