A258871 Decimal expansion of Product_{n>=1} (1+1/n^6).

2, 0, 3, 4, 7, 4, 0, 8, 3, 5, 0, 0, 9, 4, 2, 9, 0, 6, 3, 5, 8, 6, 8, 2, 0, 8, 0, 9, 6, 4, 2, 8, 5, 0, 8, 9, 7, 7, 1, 0, 9, 0, 1, 0, 0, 6, 2, 3, 9, 2, 5, 4, 6, 9, 0, 5, 5, 7, 5, 3, 9, 4, 8, 0, 4, 5, 2, 9, 8, 4, 1, 2, 0, 1, 9, 1, 5, 2, 5, 8, 4, 9, 1, 3, 5, 3, 5, 9, 8, 1, 5, 4, 9, 6, 6, 7, 0, 7, 6, 8, 6, 7, 8, 1, 3

Offset: 1

Author

Vaclav Kotesovec, Jun 13 2015

Keywords

• nonn
• cons

Comments

From Vaclav Kotesovec, Aug 30 2024: (Start)

For m>0, Product_{k>=1} (1 + m/k^6) = (cosh(Pi*m^(1/6)) - cos(sqrt(3)*Pi*m^(1/6))) * sinh(Pi*m^(1/6)) / (2*Pi^3*sqrt(m)).

If m tends to infinity, Product_{k>=1} (1 + m/k^6) ~ exp(2*Pi*m^(1/6)) / (8*Pi^3*sqrt(m)). (End)

Examples

			2.03474083500942906358682080964285089771090100623925469055753948...
		

Crossrefs

Cf. A109219, A175615, A175617, A175619, A156648, A073017, A258870, A334411.

Programs

Maple

evalf((cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3), 120);

Mathematica

RealDigits[(Cosh[Pi]-Cos[Sqrt[3]*Pi])*Sinh[Pi]/(2*Pi^3),10,120][[1]]

PARI

prodnumrat(1+x^-6, 1) \\ Charles R Greathouse IV, Feb 04 2025

Formula

Equals (cosh(Pi)-cos(sqrt(3)*Pi))*sinh(Pi)/(2*Pi^3).

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(6*j)/j)). - Vaclav Kotesovec, Mar 28 2019