A340065 - OEIS

%I #27 Jan 23 2024 08:42:10

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

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

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

%N Decimal expansion of the Product_{p>=2} 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*n))^2 = A114362(n+1)/A114363(n+1).

%C This number has decimal period length 230:

%C 1.81(0781476121562952243125904486251808972503617945007235890014471780028943

%C 5600578871201157742402315484804630969609261939218523878437047756874095

%C 5137481910274963820549927641099855282199710564399421128798842257597684

%C 51519536903039073806).

%F Equals 5005/2764 = 5*7*11*13/(2^2*691).

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

%F Equals (13/9)*A340066.

%F From _Vaclav Kotesovec_, Dec 29 2020: (Start)

%F Equals 3/2 * (Product_{p prime} (p^6+1)/(p^6-1)) * (Product_{p prime} (p^4+1)/(p^4-1)).

%F Equals 7*zeta(6)^2 / (4*zeta(12)).

%F Equals -7*binomial(12, 6) * Bernoulli(6)^2 / (8*Bernoulli(12)). (End)

%F Equals Sum_{k>=1} A005361(k)/k^2. - _Amiram Eldar_, Jan 23 2024

%e 1.8107814761215629522431259...

%t RealDigits[N[5005/2764,105]][[1]]

%o (PARI)

%o default(realprecision,105)

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

%Y Cf. A000040, A005361, A084920, A065483, A065484, A065485, A109695, A111003, A114362, A114363, A116393, A167864, A231535, A307868, A330523, A330595, A335319, A335762, A335818, A339925, A340066.

%K nonn,cons,easy

%O 1,2

%A _Artur Jasinski_, Dec 28 2020