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A146489 Decimal expansion of Product_{n>=2} (1 - 1/(n^3*(n-1))).

%I A146489 #13 Aug 06 2017 22:28:25
%S A146489 8,5,0,6,7,0,6,3,0,7,9,1,1,0,4,3,5,3,7,5,0,3,0,9,5,2,1,2,5,0,0,0,6,2,
%T A146489 3,4,9,9,9,1,5,0,5,9,8,1,9,5,4,4,2,8,3,0,6,5,6,7,6,6,0,5,6,8,2,2,9,1,
%U A146489 2,7,0,7,4,4,5,4,1,0,7,6,6,2,2,8,9,6,9,1,9,5,1,3,1,2,2,1,0,5,5,2,0,9,6,9,4
%N A146489 Decimal expansion of Product_{n>=2} (1 - 1/(n^3*(n-1))).
%C A146489 Product of Artin's constant of rank 3 and the equivalent almost-prime products.
%H A146489 Zak Seidov, <a href="/A146489/b146489.txt">Table of n, a(n) for n = 0..999</a>
%F A146489 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 = 3.
%F A146489 s*Sum_{j=1..floor(s/4)} binomial(s-3j-1, j-1)/j = A014097(s)-1.
%F A146489 Equals 1/Product_{k=1..4} Gamma(1-x_k), where x_k are the 4 roots of the polynomial x*(x+1)^3-1. [_R. J. Mathar_, Feb 20 2009]
%e A146489 0.85067063079110435... = (1 - 1/8)*(1 - 1/54)*(1 - 1/192)*(1 - 1/500)*(1 - 1/1080)*...
%p A146489 r := 3 : ni := fsolve( (n+1)^r*n-1,n,complex) : 1.0/mul(GAMMA(1-d),d=ni) ; # _R. J. Mathar_, Feb 20 2009
%t A146489 p[k_] := Gamma[1 - Root[#^4 + 3#^3 + 3#^2 + # - 1&, k]]; RealDigits[1/(p[1]*p[2]*p[3]*p[4]) // Re, 10, 105] // First (* _Jean-François Alcover_, Feb 11 2013, after _R. J. Mathar_ *)
%Y A146489 Cf. A065415.
%K A146489 nonn,cons,easy
%O A146489 0,1
%A A146489 _R. J. Mathar_, Feb 13 2009
%E A146489 Offset in b-file corrected by _N. J. A. Sloane_, Aug 31 2009
%E A146489 Data extended by _Jean-François Alcover_, Feb 11 2013