A340710 - OEIS

%I #38 Mar 03 2024 03:59:52

%S 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,

%T 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,

%U 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

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

%H Vaclav Kotesovec, <a href="/A340710/b340710.txt">Table of n, a(n) for n = 1..500</a>

%H For links see A340711.

%F D = Product_{primes p == 0 (mod 5)} (p^2+1)/(p^2-1) = 13/12.

%F E = Product_{primes p == 1 (mod 5)} (p^2+1)/(p^2-1) = A340629.

%F F = Product_{primes p == 2 (mod 5)} (p^2+1)/(p^2-1) = this constant.

%F G = Product_{primes p == 3 (mod 5)} (p^2+1)/(p^2-1) = A340711.

%F H = Product_{primes p == 4 (mod 5)} (p^2+1)/(p^2-1) = A340628.

%F D*E*F*G*H = 5/2.

%F E*F*G*H = 30/13.

%F D*E*H = sqrt(5)/2.

%F D*F*G = 13*sqrt(5)/12.

%F F*G = sqrt(5).

%F E*H = 6*sqrt(5)/13.

%F Formulas by Pascal Sebah, Jan 20 2021: (Start)

%F Let g = sqrt(Cl2(2*Pi/5)^2+Cl2(4*Pi/5)^2) = 1.0841621352693895..., where Cl2 is the Clausen function of order 2.

%F E = 15*sqrt(65)*g/(13*Pi^2).

%F H = 6*sqrt(13)*Pi^2/(195*g). (End)

%F Equals Sum_{q in A004616} 2^A001221(q)/q^2. - _R. J. Mathar_, Jan 27 2021

%e 1.7551738411687377766074721228405237...

%t (* Using Vaclav Kotesovec's function Z from A301430. *)

%t $MaxExtraPrecision = 1000; digits = 121;

%t digitize[c_] := RealDigits[Chop[N[c, digits]], 10, digits - 1][[1]];

%t digitize[1/(Z[5, 2, 4]/Z[5, 2, 2]^2)]

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

%K nonn,cons

%O 1,2

%A _Artur Jasinski_, Jan 16 2021