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A340710
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Decimal expansion of Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1).
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6
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1, 7, 5, 5, 1, 7, 3, 8, 4, 1, 1, 6, 8, 7, 3, 7, 7, 7, 6, 6, 0, 7, 4, 7, 2, 1, 2, 2, 8, 4, 0, 5, 2, 3, 7, 0, 1, 1, 1, 5, 1, 1, 8, 1, 3, 9, 4, 5, 5, 4, 3, 9, 9, 1, 5, 5, 8, 1, 7, 9, 0, 6, 2, 1, 6, 1, 7, 5, 6, 8, 6, 2, 1, 6, 4, 6, 4, 5, 1, 1, 9, 2, 7, 5, 9, 7, 9, 9, 0, 2, 4, 8, 5, 2, 5, 6, 3, 9, 7, 6, 9, 6, 3, 6, 8, 9, 5, 1, 6, 8, 2, 5, 3, 0, 2, 5, 1, 5, 1, 1
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OFFSET
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1,2
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LINKS
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FORMULA
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D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.
E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.
F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = this constant.
G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = A340711.
H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.
D*E*F*G*H = 5/2.
E*F*G*H = 30/13.
D*E*H = sqrt(5)/2.
D*F*G = 13*sqrt(5)/12.
F*G = sqrt(5).
E*H = 6*sqrt(5)/13.
Formulas by Pascal Sebah, Jan 20 2021: (Start)
Let g = sqrt(Cl2(2*Pi/5)^2+Cl2(4*Pi/5)^2) = 1.0841621352693895..., where Cl2 is the Clausen function of order 2.
E = 15*sqrt(65)*g/(13*Pi^2).
H = 6*sqrt(13)*Pi^2/(195*g). (End)
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EXAMPLE
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1.7551738411687377766074721228405237...
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MATHEMATICA
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(* Using Vaclav Kotesovec's function Z from A301430. *)
$MaxExtraPrecision = 1000; digits = 121;
digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];
digitize[1/(Z[5, 2, 4]/Z[5, 2, 2]^2)]
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CROSSREFS
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Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340711.
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KEYWORD
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AUTHOR
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STATUS
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approved
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