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%I #13 Mar 16 2021 04:06:53
%S 0,6,0,8,2,1,6,5,5,1,2,0,3,0,5,0,8,6,0,0,5,6,3,2,2,7,5,4,6,1,9,2,0,8,
%T 5,5,4,3,1,3,3,7,3,7,3,4,7,5,7,6,7,9,4,1,9,8,2,6,4,3,4,0,3,1,5,0,4,0,
%U 8,0,4,3,5,0,7,2,1,2,5,6,1,6,9,5,8,6,1,8,8,8,7,3,4,8,5,8,6,6,2,4,6,8,7,3,4,0
%N Decimal expansion of Matthews' constant C_3, an analog of Artin's constant for primitive roots.
%D Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.
%H K. R. Matthews, <a href="https://eudml.org/doc/205412">A generalisation of Artin's conjecture for primitive roots</a>, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.
%F C_3 = Product_{p prime} 1 - (p^3 - (p - 1)^3)/(p^3*(p - 1)).
%e 0.0608216551203050860056322754619208554313373734757679419826434...
%t digits = 70; $MaxExtraPrecision = 1000; m0 = 2000; dm = 200; Clear[s]; LR =
%t LinearRecurrence[{2, 2, -6, 4, -1}, {0, 6, 0, 22, 5}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> 2 m0, WorkingPrecision -> digits+10] // Exp; s[m0]; s[m = m0+dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[ s[m-dm], 10, digits][[1]], Print[m]; m = m + dm]; Join[{0}, RealDigits[ s[m], 10, digits][[1]]]
%o (PARI) prodeulerrat(1 - (p^3 - (p - 1)^3)/(p^3*(p - 1))) \\ _Amiram Eldar_, Mar 16 2021
%Y Cf. A005596, A065414, A065415, A065416, A271780, A271798.
%K nonn,cons
%O 0,2
%A _Jean-François Alcover_, Apr 16 2016
%E More digits from _Vaclav Kotesovec_, Jun 19 2020