A340794 Decimal expansion of Product_{primes p == 2 (mod 5)} p^2/(p^2-1).

1, 3, 6, 8, 5, 7, 2, 0, 5, 3, 8, 7, 6, 6, 4, 9, 0, 8, 5, 8, 6, 0, 7, 6, 3, 8, 9, 0, 4, 8, 3, 1, 0, 9, 9, 9, 0, 1, 7, 0, 2, 0, 7, 8, 2, 8, 8, 8, 5, 8, 9, 5, 2, 0, 5, 0, 0, 8, 5, 0, 4, 0, 2, 9, 5, 5, 6, 3, 3, 1, 1, 8, 8, 8, 1, 0, 5, 4, 2, 1, 2, 0, 9, 2, 1, 5, 6, 7, 7, 4, 9, 6, 0, 8, 0, 9, 7, 3, 8, 1, 1, 9, 4, 4, 2, 9, 3, 2, 4, 3, 5, 1, 5, 4, 0, 9, 3, 2, 2, 6

Offset: 1

Author

Artur Jasinski, Jan 21 2021

Keywords

• nonn
• cons

Examples

			1.36857205387664908586076389048310999017020782888589520500850402955633118881...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..501
• For links see A340711.

Crossrefs

Cf. A004616, A175646, A175647, A248930, A248938, A301429, A333240, A334826, A335963, A340127, A340576, A340577, A340578, A340628, A340629, A340665, A340710, 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[Z[5, 2, 2]]

Formula

I = Product_{primes p == 0 (mod 5)} p^2/(p^2-1) = 25/24.

J = Product_{primes p == 1 (mod 5)} p^2/(p^2-1) = A340004.

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

L = Product_{primes p == 3 (mod 5)} p^2/(p^2-1) = A340665.

M = Product_{primes p == 4 (mod 5)} p^2/(p^2-1) = A340127.

I*J*K*L*M = Pi^2/6 = zeta(2).

J*K*L*M = 4*Pi^2/25.

M = (Pi/2)*C(5,4)^(-2)*exp(-gamma/2)*sqrt(3/log(2+sqrt(5))), where gamma is the Euler-Mascheroni constant A001620 and C(5,4) is the Mertens constant = 1.29936454791497798816084...

Equals Sum_{k>=1} 1/A004616(k)^2. - Amiram Eldar, Jan 24 2021