OFFSET
0,1
COMMENTS
Product of Artin's constant of rank 2 and the equivalent almost-prime products.
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.
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]
EXAMPLE
0.6739173633763... = (1 - 1/4)*(1 - 1/18)*(1 - 1/48)*(1 - 1/100)*...
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 *)
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Feb 13 2009
EXTENSIONS
More terms from Jean-François Alcover, Feb 12 2013
STATUS
approved