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%I #24 Aug 05 2023 13:10:18
%S 5,7,5,9,5,9,9,6,8,8,9,2,9,4,5,4,3,9,6,4,3,1,6,3,3,7,5,4,9,2,4,9,6,6,
%T 9,2,5,0,6,5,1,3,9,6,7,1,7,6,4,9,2,3,6,3,6,0,0,6,4,0,7,9,8,6,6,5,3,7,
%U 2,5,5,1,6,9,8,8,6,8,5,2,8,4,3,6,4,0,9,8,7,2,0,9,1,7,2,6,1,8
%N Decimal expansion of Product_{p prime} (1 - p/(p^3-1)).
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 106.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 155.
%H Yilan Hu and Carl Pomerance, <a href="http://dx.doi.org/10.2140/involve.2012.5.229">The average order of elements in the multiplicative group of a finite field</a>, involve, Vol. 5 (2012), No. 2, 229-236. See p. 8.
%H Sungjin Kim, <a href="https://doi.org/10.1016/j.jnt.2016.05.019">Average results on the order of a modulo p</a>, Journal of Number Theory, Vol. 169 (2016), pp. 353-368; <a href="http://arxiv.org/abs/1509.01752">arXiv preprint</a>, arXiv:1509.01752 [math.NT], 2015.
%H Pär Kurlberg and Carl Pomerance, <a href="http://dx.doi.org/10.2140/ant.2013.7.981">On a problem of Arnold: the average multiplicative order of a given integer</a>, Algebra and Number Theory, Vol. 7, No. 4 (2013), pp. 981-999; <a href="https://math.dartmouth.edu/~carlp/arnoldfinal.pdf">alternative link</a>.
%H Pieter Moree and Peter Stevenhagen, <a href="https://doi.org/10.1006/jnth.2000.2547">A two-variable Artin conjecture</a>, Journal of Number Theory, Vol. 85, No. 2 (2000), pp. 291-304; <a href="https://arxiv.org/abs/math/9912250">arXiv preprint</a>, arXiv:math/9912250 [math.NT], 1999.
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%H P. J. Stephens, <a href="https://doi.org/10.1112/S0025579300008159">An average result for Artin's conjecture</a>, Mathematika, Vol. 16, No. 2 (1969), pp. 178-188.
%H P. J. Stephens, <a href="http://dx.doi.org/10.1016/0022-314X(76)90010-X">Prime divisors of second-order linear recurrences. I,</a>, Journal of Number Theory, Vol. 8, No. 3 (1976), pp. 313-332.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/StephensConstant.html">Stephens' Constant</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Stephens%27_constant">Stephens' constant</a>.
%e 0.57595996889294543964316337549249669...
%t $MaxExtraPrecision = 100; m0 = 200; dm = 200; digits = 101; Clear[f]; f[m_] := f[m] = (slog = Normal[Series[Log[1 - p/(p^3 - 1)], {p, Infinity, m}]]; Exp[slog] /. Power[p, n_] -> PrimeZetaP[-n] // N[#, digits+10]&); f[m = m0]; Print[m, " ", f[m]]; f[m = m + dm]; While[Print[m, " ", f[m]]; RealDigits[f[m], 10, digits+5] != RealDigits[f[m - dm], 10, digits+5], m = m + dm];RealDigits[f[m], 10, digits] // First (* _Jean-François Alcover_, Sep 15 2015 *)
%o (PARI) prodeulerrat(1 - p/(p^3-1)) \\ _Amiram Eldar_, Mar 17 2021
%Y Cf. A078079.
%K cons,nonn
%O 0,1
%A _N. J. A. Sloane_, Nov 19 2001
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