A175616 Decimal expansion of product_{n>=2} (1-n^(-5)).
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, 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, 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
Offset: 0
Examples
0.96325656175755909737304603488397519554352075785342263739516...
Links
- E. Weisstein, Infinite Product, Mathworld.
Programs
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Mathematica
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 *)
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PARI
exp(suminf(j=1, (1 - zeta(5*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020
Formula
Equals product_{t=1..4} 1/Gamma(2-exp(2*Pi*i*t/5)), where i is the imaginary unit.
Equals exp(Sum_{j>=1} (1 - zeta(5*j))/j). - Vaclav Kotesovec, Apr 27 2020
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