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A065484 Decimal expansion of Product_{p prime >= 2} (1 + p/((p-1)^2*(p+1))).

%I A065484 #34 Feb 16 2025 08:32:45
%S A065484 2,2,0,3,8,5,6,5,9,6,4,3,7,8,5,9,7,8,7,8,7,2,8,2,8,3,1,6,4,8,0,0,8,9,
%T A065484 6,6,2,5,6,7,1,7,3,1,9,3,7,8,7,8,5,8,6,3,4,1,7,0,4,9,5,5,4,4,8,7,1,6,
%U A065484 6,8,8,6,8,1,1,8,5,2,6,9,5,4,9,7,5,7,2,6,6,0,4,1,9,0,1,3,9,5,6
%N A065484 Decimal expansion of Product_{p prime >= 2} (1 + p/((p-1)^2*(p+1))).
%C A065484 Decimal expansion of totient constant. - _Eric W. Weisstein_, Apr 20 2006
%H A065484 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. 86.
%H A065484 G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%H A065484 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/TotientSummatoryFunction.html">Totient Summatory Function</a>.
%F A065484 Equals Pi^2 * A065483 / 6.
%F A065484 Also defined as: Sum_{m>=1} 1/(m*A000010(m)). See Weisstein link.
%F A065484 Equals 5 * Sum_{m>=1} (-1)^(m+1)/(m*A000010(m)). - _Amiram Eldar_, Nov 21 2022
%e A065484 2.203856596437859787872828316480...
%t A065484 $MaxExtraPrecision = 500; digits = 99; terms = 500; P[n_] := PrimeZetaP[n];
%t A065484 LR = Join[{0, 0, 0}, LinearRecurrence[{2, -1, -1, 1}, {3, 4, 5, 3}, terms + 10]]; r[n_Integer] := LR[[n]];  (Pi^2/6)*Exp[NSum[r[n]*P[n - 1]/(n - 1), {n, 3, terms}, NSumTerms -> terms, WorkingPrecision -> digits + 10]  ] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Apr 18 2016 *)
%o A065484 (PARI) prodeulerrat(1 + p/((p-1)^2*(p+1))) \\ _Hugo Pfoertner_, Jun 02 2020
%Y A065484 Cf. A000010, A002618, A065483, A077387.
%K A065484 cons,nonn
%O A065484 1,1
%A A065484 _N. J. A. Sloane_, Nov 19 2001, Aug 09 2010