A146492 - OEIS

%I #12 Aug 06 2017 22:28:43

%S 9,2,9,8,3,8,4,7,3,9,5,4,3,4,6,8,5,2,2,3,8,3,1,8,4,6,9,5,3,4,5,5,3,5,

%T 4,8,9,4,4,9,0,8,3,0,5,4,8,2,2,5,3,6,3,5,2,3,6,7,5,7,4,9,7,0,6,9,7,3,

%U 5,3,7,8,0,0,2,2,5,8,0,8,2,2,3,2,2,2,3,4,5,7,6,6,4,0,2,7,0,3,1,2,0,2,4,1,5

%N Decimal expansion of Product_{n>=2} (1 - 1/(n^4*(n-1))).

%C Product of Artin's constant of rank 4 and the equivalent almost-prime products.

%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 = 4.

%F s*Sum_{j=1..floor(s/5)} binomial(s-4j-1, j-1)/j = A058368(s)-1.

%F Equals 1/Product_{k=1..5} Gamma(1-x_k), where x_k are the 5 roots of the polynomial x*(x+1)^4-1. [_R. J. Mathar_, Feb 20 2009]

%e 0.9298384739543468522383.. = (1-1/16)*(1-1/162)*(1-1/768)*(1-1/2500)*..

%p r := 4 : 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[-5 + 11# - 6#^2 + #^3 & , k]]; RealDigits[Cosh[(Sqrt[3] Pi)/2]/(Pi g[1] g[2] g[3]) // Re, 10, 105] // First (* _Jean-François Alcover_, Feb 12 2013 *)

%t RealDigits[Re[N[Product[1 - 1/(n^4 (n - 1)), {n, 2, Infinity}], 110]]] (* _Bruno Berselli_, Apr 02 2013 *)

%Y Cf. A065416.

%K nonn,cons,easy

%O 0,1

%A _R. J. Mathar_, Feb 13 2009

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