A340065 Decimal expansion of the Product_{p>=2} 1+p^2/((p-1)^2*(p+1)^2) where p are successive prime numbers A000040.

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, 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, 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

Offset: 1

Author

Artur Jasinski, Dec 28 2020

Keywords

• nonn
• cons
• easy

Comments

This is a rational number.

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),

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

This number has decimal period length 230:

1.81(0781476121562952243125904486251808972503617945007235890014471780028943

5600578871201157742402315484804630969609261939218523878437047756874095

5137481910274963820549927641099855282199710564399421128798842257597684

51519536903039073806).

Examples

			1.8107814761215629522431259...
		

Crossrefs

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

Programs

Mathematica

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

PARI

default(realprecision,105)
prodeulerrat(1+p^2/((p-1)^2*(p+1)^2))

Formula

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

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

Equals (13/9)*A340066.

From Vaclav Kotesovec, Dec 29 2020: (Start)

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

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

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

Equals Sum_{k>=1} A005361(k)/k^2. - Amiram Eldar, Jan 23 2024