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

%I #13 Aug 06 2017 22:28:20

%S 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,

%T 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,

%U 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

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

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

%H R. J. Mathar, <a href="http://arxiv.org/abs/0903.2514">Hardy-Littlewood constants embedded into infinite products over all positive integers</a>, arXiv:0903.2514 [math.NT], table 3, first line with r=2.

%F 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.

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

%F 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]

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

%p 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

%t 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 *)

%Y Cf. A065414.

%K nonn,cons,easy

%O 0,1

%A _R. J. Mathar_, Feb 13 2009

%E More terms from _Jean-François Alcover_, Feb 12 2013