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%I #13 Dec 15 2020 07:05:15
%S 9,6,3,2,5,6,5,6,1,7,5,7,5,5,9,0,9,7,3,7,3,0,4,6,0,3,4,8,8,3,9,7,5,1,
%T 9,5,5,4,3,5,2,0,7,5,7,8,5,3,4,2,2,6,3,7,3,9,5,1,6,8,8,5,0,4,2,7,6,9,
%U 4,4,2,1,8,8,7,6,7,8,1,3,0,4,6,3,6,3,5,8,0,4,6,8,6,0,9,7,9,6,9,8,7,0,9,6,8
%N Decimal expansion of product_{n>=2} (1-n^(-5)).
%H E. Weisstein, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>, Mathworld.
%F Equals product_{t=1..4} 1/Gamma(2-exp(2*Pi*i*t/5)), where i is the imaginary unit.
%F Equals exp(Sum_{j>=1} (1 - zeta(5*j))/j). - _Vaclav Kotesovec_, Apr 27 2020
%F Equals 1/(Gamma(2 + phi/2 - i*(5^(1/4) / (2*sqrt(phi)))) * Gamma(2 + phi/2 + i*(5^(1/4) / (2*sqrt(phi)))) * Gamma(2 - 1/(2*phi) - i*5^(1/4)*(sqrt(phi)/2)) * Gamma(2 - 1/(2*phi) + i*5^(1/4)*(sqrt(phi)/2))), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio and i is the imaginary unit. - _Vaclav Kotesovec_, Dec 15 2020
%e 0.96325656175755909737304603488397519554352075785342263739516...
%t g[k_] := Gamma[Root[1 - # + #^2 - #^3 + #^4 & , k]]; RealDigits[ 1/(5*g[1]*g[2]*g[3]*g[4]) // Re, 10, 105] // First (* _Jean-François Alcover_, Feb 12 2013 *)
%o (PARI) exp(suminf(j=1, (1 - zeta(5*j))/j)) \\ _Vaclav Kotesovec_, Apr 27 2020
%Y Cf. A109219, A175615, A144664.
%K cons,easy,nonn
%O 0,1
%A _R. J. Mathar_, Jul 26 2010