A340066 - OEIS

%I #17 Jan 01 2021 14:34:09

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

%T 5,6,0,0,5,7,8,8,7,1,2,0,1,1,5,7,7,4,2,4,0,2,3,1,5,4,8,4,8,0,4,6,3,0,

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

%N Decimal expansion of the Product_{p>=3} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

%C This is a rational number.

%C This constant does not belong to the infinite series of prime number products of the form: Product_{p>=2} (p^(2*n)-1)/(p^(2*n)+1),

%C which are rational numbers equal to zeta(4*n)/zeta^2(2*n) = A114362(n+1)/A114363(n+1).

%C This number has decimal period length 230:

%C 1.25(3617945007235890014471780028943560057887120115774240231548480463096960

%C 9261939218523878437047756874095513748191027496382054992764109985528219

%C 9710564399421128798842257597684515195369030390738060781476121562952243

%C 12590448625180897250).

%F Equals 3465/2764 = 3^2*5*7*11/(2^2*691).

%F Equals Product_{n>=2} 1+A000040(n)^2/A084920(n)^2.

%F Equals (9/13)*A340065.

%e 1.25361794500723589001447178...

%t RealDigits[N[3465/2764, 105]][[1]]

%o (PARI)

%o default(realprecision, 105)

%o prodeulerrat(1+p^2/((p-1)^2*(p+1)^2),1,3)

%Y Cf. A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523,  A330595, A335319, A335762, A335818, A339925, A340065.

%K nonn,cons

%O 1,2

%A _Artur Jasinski_, Dec 28 2020