A175619 -id:A175619 - cp's OEIS frontend

A175615 Decimal expansion of sinh(Pi)/(4*Pi).

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

Offset: 0

R. J. Mathar, Jul 26 2010

			0.91901947759...

R. J. Mathar, Hardy-Littlewood constants embedded into infinite products..., arXiv:0903.2514.

Cf. A109219, A156648, A144663, A175617, A175619.

Maple
```
sinh(Pi)/4/Pi; evalf(%) ;
```

RealDigits[Sinh[Pi]/(4Pi),10,120][[1]] (* Harvey P. Dale, Feb 11 2023 *)

exp(suminf(j=1, (1 - zeta(4*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020

Equals product_{n >= 2} (1-n^(-4)).
Equals A156648/4.
Equals exp(Sum_{j>=1} (1 - zeta(4*j))/j). - Vaclav Kotesovec, Apr 27 2020
Equals 1/(2*Gamma(2-i)*Gamma(2+i)). - Amiram Eldar, May 28 2021

A109219 Decimal expansion of Product_{n >= 2} 1-n^(-3).

Original entry on oeis.org

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

Offset: 0

Zak Seidov, Apr 17 2006

The physical applications of this kind of product (with s<0) can be found in the Klauder et al. reference. - Karol A. Penson, Feb 24 2006

			0.809396597366290109578680478726382119372787648261130...

J. R. Klauder, K. A. Penson and J.-M. Sixdeniers, Constructing coherent states through solutions of Stieltjes and Hausdorff moment problems, Physical Review A, Vol. 64, p. 013817 (2001).

Cf. A073017, A175615, A175617, A175619, A258870.

RealDigits[N[Cosh[(Sqrt[3]*Pi)/2]/(3*Pi), 150]] (* T. D. Noe, Apr 24 2006 *)

exp(suminf(j=1, (1 - zeta(3*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020

prodnumrat(1-1/x^3,2) \\ Charles R Greathouse IV, Feb 03 2025

Equals cosh((sqrt(3)*Pi)/2)/(3*Pi).
Product_{n >= 2} (1 - 1/n^p) simplifies, if p is odd, to 1/(p * Product_{j=1..p-1} Gamma(-(-1)^(j*(1 + 1/p)))) and, if p is even, to the elementary (Product_{j=1..p/2-1} sin(Pi*(-1)^(2*j/p))/(Pi*i)) / p. - David W. Cantrell, Feb 24 2006
Equals exp(Sum_{j>=1} (1 - zeta(3*j))/j). - Vaclav Kotesovec, Apr 27 2020
Equals 1/(Gamma((5-i*sqrt(3))/2)*Gamma((5+i*sqrt(3))/2)). - Amiram Eldar, Sep 01 2020

Corrected and extended by T. D. Noe, Apr 24 2006

A175617 Decimal expansion of product_{n>=2} (1-n^(-6)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 26 2010

			0.9826842777...

E. Weisstein, Infinite Product, Mathworld.

Cf. A109219, A175615, A175619, A144665, A258871.

cosh(Pi*sqrt(3)/2)^2/6/Pi^2 ; evalf(%) ;

RealDigits[(1 + Cosh[Sqrt[3]*Pi])/(12*Pi^2), 10, 105] // First (* Jean-François Alcover, Feb 12 2013 *)

exp(suminf(j=1, (1 - zeta(6*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020

Equals product_{t=1..5} 1/Gamma(2-exp(Pi*i*t/3)), where i is the imaginary unit and Pi/3 = A019670.

Equals exp(Sum_{j>=1} (1 - zeta(6*j))/j). - Vaclav Kotesovec, Apr 27 2020

A258870 Decimal expansion of Product_{n>=1} (1+1/n^4).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Jun 13 2015

For m>0, Product_{k>=1} (1 + m/k^4) = (cosh(sqrt(2)*Pi*m^(1/4)) - cos(sqrt(2)*Pi*m^(1/4))) / (2*Pi^2*sqrt(m)). - Vaclav Kotesovec, Aug 30 2024

			2.16736062588226195190023136684702744182161317296349850975623259988...

Cf. A109219, A175615, A175617, A175619, A156648, A073017, A258871, A334411.

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

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

exp(sumalt(j=1, -(-1)^j*zeta(4*j)/j)) \\ Vaclav Kotesovec, Dec 15 2020

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

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(4*j)/j)). - Vaclav Kotesovec, Mar 28 2019

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

Original entry on oeis.org

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

Vaclav Kotesovec, Jun 13 2015

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)

			2.03474083500942906358682080964285089771090100623925469055753948...

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

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

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

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

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

A334411 Decimal expansion of Product_{k>=1} (1 + 1/k^8).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Apr 27 2020

From Vaclav Kotesovec, Aug 30 2024: (Start)
For m>0, Product_{k>=1} (1 + m/k^8) = (cosh(Pi*sqrt(2 - sqrt(2))*m^(1/8)) - cos(Pi*sqrt(2 + sqrt(2))*m^(1/8))) * (cosh(Pi*sqrt(2 + sqrt(2))*m^(1/8)) - cos(Pi*sqrt(2 - sqrt(2))*m^(1/8)))/(4*sqrt(m)*Pi^4).
If m tends to infinity, Product_{k>=1} (1 + m/k^8) ~ exp(Pi*sqrt(2*(2 + sqrt(2)))*m^(1/8)) / (16*Pi^4*sqrt(m)).
In general, if m tends to infinity and v > 2, Product_{k>=1} (1 + m/k^v) ~ exp(Pi*m^(1/v)/sin(Pi/v)) / ((2*Pi)^(v/2)*sqrt(m)). (End)

			2.00815605499274531514903948232341369211953215983095097877074299617422...

Cf. A156648, A073017, A258870, A307216, A258871.

Cf. A109219, A175615, A175616, A175617, A175619.

evalf(Product(1 + 1/j^8, j = 1..infinity), 120);

RealDigits[Chop[N[Product[(1 + 1/n^8), {n, 1, Infinity}], 120]]][[1]]

default(realprecision, 120); exp(sumalt(j=1, -(-1)^j*zeta(8*j)/j))

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(8*j)/j)).

Equals (cos(sqrt(4 - 2*sqrt(2))*Pi) + cos(sqrt(4 + 2*sqrt(2))*Pi) + cosh(sqrt(4 - 2*sqrt(2))*Pi) + cosh(sqrt(4 + 2*sqrt(2))*Pi) - 2*cos(sqrt(2 - sqrt(2))*Pi) * cosh(sqrt(2 - sqrt(2))*Pi) - 2*cos(sqrt(2 + sqrt(2))*Pi) * cosh(sqrt(2 + sqrt(2))*Pi)) / (8*Pi^4).

A144667 Decimal expansion of product_{n=2..infinity} (n^8-1)/(n^8+1).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Feb 01 2009

			0.9918784076582948169664...

J. Borwein et al., Experimentation in Mathematics, 2004, section 1.2.
E. W. Weisstein, Infinite Product, MathWorld.

Cf. A090986.

RealDigits[-1/4*Pi*Csc[(-1)^(1/8)*Pi]*Csc[(-1)^(3/8)*Pi]*Csc[(-1)^(5/8)*Pi] * Csc[(-1)^(7/8)*Pi]*Sin[(-1)^(1/4)*Pi]*Sin[(-1)^(3/4)*Pi]*Sinh[Pi] // Re, 10, 105] // First (* Jean-François Alcover, Feb 11 2013 *)

Equals 2*A175619/A334411. - Vaclav Kotesovec, Apr 27 2020

More terms from Jean-François Alcover, Feb 11 2013

A339745 Decimal expansion of Product_{n>=2} (1 - n^(-10)).

Original entry on oeis.org

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

Offset: 0

Vaclav Kotesovec, Dec 15 2020

			0.99900544248098947527378453585422724586059097385364736908229...

Cf. A109219, A175615, A175616, A175617, A175618, A175619.

evalf((cosh(sqrt((5 - sqrt(5))/2)*Pi) + sin(sqrt(5)*Pi/2)) * (cosh(sqrt((5 + sqrt(5))/2)*Pi) - sin(sqrt(5)*Pi/2)) / (40*Pi^4), 100);

RealDigits[(Cosh[Sqrt[(5 - Sqrt[5])/2]*Pi] + Sin[Sqrt[5]*Pi/2]) * (Cosh[Sqrt[(5 + Sqrt[5])/2]*Pi] - Sin[Sqrt[5]*Pi/2]) / (40*Pi^4), 10, 100][[1]]

PARI
```
exp(suminf(j=1, (1 - zeta(10*j))/j))
```

prodinf(n=2, 1-1/n^10) \\ Michel Marcus, Dec 15 2020

Equals (cosh(sqrt((5 - sqrt(5))/2)*Pi) + sin(sqrt(5)*Pi/2)) * (cosh(sqrt((5 + sqrt(5))/2)*Pi) - sin(sqrt(5)*Pi/2)) / (40*Pi^4).

Equals exp(Sum_{j>=1} (1 - zeta(10*j))/j).

A346745 Decimal expansion of Product_{k>=2} (1 - 1/k^12).

Original entry on oeis.org

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

Offset: 0

Sean A. Irvine, Jul 31 2021

			0.999753913921893256003448570641909727180...

Michael I. Shamos, Shamos's catalog of the real numbers (2011).

Cf. A109219, A175615, A175616, A175617, A175618, A175619, A339745.

evalf(sinh(Pi) * cosh(Pi*sqrt(3)/2)^2 * (cosh(Pi) - cos(Pi*sqrt(3))) / (24*Pi^5), 120); # Vaclav Kotesovec, Aug 01 2021

RealDigits[Sinh[Pi]*Cosh[Pi*Sqrt[3]/2]^2*(Cosh[Pi] - Cos[Pi*Sqrt[3]])/(24*Pi^5), 10, 120][[1]] (* Amiram Eldar, Jun 12 2023 *)

exp(suminf(j=1, (1 - zeta(12*j))/j)) \\ Vaclav Kotesovec, Aug 01 2021

Equals sinh(Pi) * cosh(Pi*sqrt(3)/2)^2 * (cosh(Pi) - cos(Pi*sqrt(3))) / (24*Pi^5).

Equals exp(Sum_{j>=1} (1 - zeta(12*j))/j). - Vaclav Kotesovec, Aug 01 2021

cp's OEIS Frontend

A175615 Decimal expansion of sinh(Pi)/(4*Pi).

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A109219 Decimal expansion of Product_{n >= 2} 1-n^(-3).

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A175617 Decimal expansion of product_{n>=2} (1-n^(-6)).

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A258870 Decimal expansion of Product_{n>=1} (1+1/n^4).

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

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A334411 Decimal expansion of Product_{k>=1} (1 + 1/k^8).

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A144667 Decimal expansion of product_{n=2..infinity} (n^8-1)/(n^8+1).

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