A340710 Decimal expansion of Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1).

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

Offset: 1

Author

Artur Jasinski, Jan 16 2021

Keywords

• nonn
• cons

Examples

			1.7551738411687377766074721228405237...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..500
• For links see A340711.

Crossrefs

Cf. A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340576, A340577, A340578, A340628, A340629, A340004, A340127, A340711.

Programs

Mathematica

(* 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)]

Formula

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)

Equals Sum_{q in A004616} 2^A001221(q)/q^2. - R. J. Mathar, Jan 27 2021