A175615 -id:A175615 - cp's OEIS frontend

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

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

A073017 Decimal expansion of the Product_{n>=1} (1 + 1/n^3).

Original entry on oeis.org

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

Offset: 1

Robert G. Wilson v, Aug 03 2002

Let X_1, X_2, ... be a sequence of independent Bernoulli trials with probability of success 1/n^3. Let Y be the position of the last success in the sequence. 1.428189... is the expected value of Y. - Geoffrey Critzer, Aug 19 2019

If m tends to infinity, Product_{k>=1} (1 + m/k^3) ~ exp(2*Pi*m^(1/3)/sqrt(3)) / (2^(3/2)*Pi^(3/2)*sqrt(m)). - Vaclav Kotesovec, Aug 30 2024

			2.42818979209887032873604143617914635811836294478339049763...

Simon Plouffe, product(1+1/n**3,n=1..infinity).
S. Ramanujan, Notebook entry.

Cf. A000984, A006014, A109219, A175615, A175617, A156648, A258870, A258871, A334411.

RealDigits[ Cosh[Sqrt[3]*Pi/2]/Pi, 10, 105][[1]] (* Jean-François Alcover, Nov 18 2015 *)

Equals cosh(1/2 * sqrt(3) * Pi)/Pi.
Equals exp(Sum_{j>=1} (-(-1)^j*zeta(3*j)/j)). - Vaclav Kotesovec, Mar 28 2019
Equals Product_{n>=1} (1 + 1/(n^2 + n)). - Amiram Eldar, Sep 01 2020
Equals 3*Product_{n >= 2} (1-n^(-3)) = 3*A109219. - Robert FERREOL, Oct 06 2021

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

A175619 Decimal expansion of Product_{n>=2} (1-n^(-8)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 26 2010

			0.9959233150... = (255/256)*(6560/6561)*(65535/65536)*...

E. Weisstein, Infinite Product, Mathworld.

Cf. A109219, A175615, A175617, A144667, A334411.

t := Pi/sqrt(2) ; sinh(Pi)*((sin(t)*cosh(t))^2+(cos(t)*sinh(t))^2)/8/Pi^3 ; evalf(%) ;

RealDigits[ -Sin[(-1)^(1/4)*Pi]*Sin[(-1)^(3/4)*Pi]*Sinh[Pi] / (8*Pi^3) // Re, 10, 105] // First(* Jean-François Alcover, Feb 12 2013 *)

exp(suminf(j=1, (1 - zeta(8*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020

prodnumrat(1-x^-8, 2) \\ Charles R Greathouse IV, Feb 04 2025

Equals (cosh(sqrt(2)*Pi) - cos(sqrt(2)*Pi)) * sinh(Pi) / (16*Pi^3). - Vaclav Kotesovec, Apr 27 2020

Equals exp(Sum_{j>=1} (1 - zeta(8*j))/j). - Vaclav Kotesovec, Apr 27 2020

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).

A175616 Decimal expansion of product_{n>=2} (1-n^(-5)).

Original entry on oeis.org

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

R. J. Mathar, Jul 26 2010

			0.96325656175755909737304603488397519554352075785342263739516...

E. Weisstein, Infinite Product, Mathworld.

Cf. A109219, A175615, A144664.

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 *)

exp(suminf(j=1, (1 - zeta(5*j))/j)) \\ Vaclav Kotesovec, Apr 27 2020

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

A175618 Decimal expansion of product_{n>=2} (1-n^(-7)).

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 26 2010

			0.99165495...

E. Weisstein, Infinite Product, Mathworld.

Cf. A109219, A175615, A144666.

N[1/(7*Product[ Gamma[(-1)^(8*k/7 + 1)], {k, 1, 6}]), 105] // Re // RealDigits // First (* Jean-François Alcover, Feb 05 2013 *)

exp(suminf(j=1, (1 - zeta(7*j))/j)) \\ Vaclav Kotesovec, Dec 15 2020

Equals 1/product_{t=1..6} Gamma(2-exp(2*Pi*i*t/7)), where i is the imaginary unit and 2*Pi/7 = A019695.

Equals exp(Sum_{j>=1} (1 - zeta(7*j))/j). - Vaclav Kotesovec, Dec 15 2020

A330864 Decimal expansion of sinh(Pi/2)/2.

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 28 2020

This constant is transcendental.

			(1 + 1/2^2) * (1 - 1/3^2) * (1 + 1/4^2) * (1 - 1/5^2) * (1 + 1/6^2) * ... = (e^(Pi/2) - e^(-Pi/2))/4 = 1.15064945115364743673152001...

Index entries for transcendental numbers

Cf. A042972, A049006, A090986, A156648, A175615, A308715, A308716, A308718, A323983, A330865, A334401.

Mathematica
```
RealDigits[Sinh[Pi/2]/2, 10, 110] [[1]]
```

sinh(Pi/2)/2 \\ Michel Marcus, Apr 28 2020

Equals Sum_{k>=1} Pi^(2*k-1)/(4^k*(2*k-1)!).
Equals Product_{k>=2} (1 + (-1)^k/k^2).
Equals (i^(-i) - i^i)/4, where i is the imaginary unit.

cp's OEIS Frontend

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

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A073017 Decimal expansion of the Product_{n>=1} (1 + 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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A175619 Decimal expansion of Product_{n>=2} (1-n^(-8)).

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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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