A146485 Decimal expansion of Product_{n>=2} (1 - 1/(n^2*(n-1))).

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

Offset: 0

Author

R. J. Mathar, Feb 13 2009

Keywords

• nonn
• cons
• easy

Comments

Product of Artin's constant of rank 2 and the equivalent almost-prime products.

Examples

			0.6739173633763... = (1 - 1/4)*(1 - 1/18)*(1 - 1/48)*(1 - 1/100)*...
		

Links

• R. J. Mathar, Hardy-Littlewood constants embedded into infinite products over all positive integers, arXiv:0903.2514 [math.NT], table 3, first line with r=2.

Crossrefs

Cf. A065414.

Programs

Maple

r := 2 : ni := fsolve( (n+1)^r*n-1,n,complex) : 1.0/mul(GAMMA(1-d),d=ni) ; # R. J. Mathar, Feb 20 2009

Mathematica

g[k_] := Gamma[Root[-3 + 8# - 5#^2 + #^3 & , k]]; RealDigits[1/(g[1]*g[2]*g[3]) // Re, 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

Formula

The logarithm is -Sum_{s>=2} Sum_{j=1..floor(s/(1+r))} binomial(s-r*j-1, j-1)*(1-Zeta(s))/j at r = 2.

s*Sum_{j=1..floor(s/3)} binomial(s-2j-1, j-1)/j = A001609(s)-1.

Equals 1/Product_{k=1..3} Gamma(1-x_k), where x_k are the 3 roots of the polynomial x*(x+1)^2-1. [R. J. Mathar, Feb 20 2009]

Extensions

More terms from Jean-François Alcover, Feb 12 2013