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%I A271877 #13 Mar 16 2021 04:06:44
%S A271877 0,2,6,1,0,7,4,4,6,3,1,4,9,1,7,7,0,8,0,8,3,2,4,9,3,9,4,3,1,3,8,2,1,4,
%T A271877 6,7,2,5,5,6,2,6,6,7,3,6,4,0,5,5,3,8,0,4,5,2,7,6,1,1,7,3,3,7,1,0,2,4,
%U A271877 9,8,2,0,0,5,6,5,8,7,0,1,4,0,9,9,6,8,4,7,0,4,8,1,5,1,1,5,2,2,6,0,3,8,6,9,4,0
%N A271877 Decimal expansion of Matthews' constant C_4, an analog of Artin's constant for primitive roots.
%D A271877 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.
%H A271877 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 A271877 C_4 = Product_{p prime} 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)).
%e A271877 0.026107446314917708083...
%t A271877 $MaxExtraPrecision = 2000; LR = LinearRecurrence[{2, 3, -10, 10, -5, 1}, {0, -8, 6, -40, 35, -194}, 10^4]; r[n_Integer] := LR[[n]]; NSum[r[n] PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2000, WorkingPrecision -> 300, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, 20]& // First // Prepend[#, 0]&
%t A271877 $MaxExtraPrecision = 1000; Clear[f]; f[p_] := 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)); Do[c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 105]]], {m, 100, 1000, 100}] (* _Vaclav Kotesovec_, Jun 19 2020 *)
%o A271877 (PARI) prodeulerrat(1 - (p^4 - (p - 1)^4)/(p^4*(p - 1))) \\ _Amiram Eldar_, Mar 16 2021
%Y A271877 Cf. A005596, A065414, A065415, A065416, A271780, A271798, A271869.
%K A271877 nonn,cons
%O A271877 0,2
%A A271877 _Jean-François Alcover_, Apr 16 2016
%E A271877 More digits from _Vaclav Kotesovec_, Jun 19 2020