A258871 -id:A258871 - cp's OEIS frontend

A156648 Decimal expansion of Product_{k>=1} (1 + 1/k^2).

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

Offset: 1

R. J. Mathar, Feb 12 2009

Consider the value at s = 2 of the partition zeta functions zeta_{type}(s), where the defining sum runs over partitions into 'type' parts, where 'type' is 'even', 'prime' or 'distinct'. (For the precise definitions see R. Schneider's dissertation.) Then
  zeta_{even}(2) = Pi/2 = A019669;
  zeta_{prime}(2) = Pi^2/6 = A013661;
  zeta_{distinct}(2) = sinh(Pi)/Pi, this constant. - Peter Luschny, Aug 11 2021
For m>0, Product_{k>=1} (1 + m/k^2) = sinh(Pi*sqrt(m)) / (Pi*sqrt(m)). - Vaclav Kotesovec, Aug 30 2024

			3.676077910374977720695697492028260666507156346827630277478003593557447324111... = (1+1)*(1+1/4)*(1+1/9)*(1+1/16)*(1+1/25)*...

Reinhold Remmert, Classical topics in complex function theory, Vol. 172 of Graduate Texts in Mathematics, p. 12, Springer, 1997.

Robert Schneider, Eulerian series, zeta functions and the arithmetic of partitions, arXiv:2008.04243 [math.NT], 2020.

Square root of A084243.

Cf. A084243, A073017, A258870, A258871, A334411, A019669, A013661.

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

RealDigits[Sinh[Pi]/Pi, 10, 111][[1]] (* or *)
RealDigits[Re[1/(I!*(-I)!)], 10, 111][[1]] (* Robert G. Wilson v, Feb 23 2015 *)

sinh(Pi)/Pi \\ Charles R Greathouse IV, Dec 16 2013

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

Equals sinh(Pi)/Pi.
Equals 1/A090986. - R. J. Mathar, Mar 05 2009
Binomial(2, 1+i) = 1/(i!*(-i)!) (where x! means Gamma(x+1)). - Robert G. Wilson v, Feb 23 2015
Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(2*j)/j)). - Vaclav Kotesovec, Mar 28 2019
Equals Product_{k>=1} (1+2/(k*(k+2))). - Amiram Eldar, Aug 16 2020

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

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).

A375843 a(n) = Product_{k=0..n} (k^6 + n).

Original entry on oeis.org

0, 2, 396, 588528, 4087208000, 97390176449400, 6465177278417919600, 1030450155933504769261568, 352805275175791344554903371776, 238031797291547406166218644352900000, 295416986525310718941438520613968960376000

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

Cf. A258871, A354054.

Cf. A375839, A375840, A375841, A375842, A375844, A375845.

Table[Product[k^6 + n, {k, 0, n}], {n, 0, 15}]

a(n) ~ n^(6*n + 7/2) / exp(6*n - 2*Pi*n^(1/6)).

A307216 Decimal expansion of Product_{k>=1} (1 + 1/k^5).

Original entry on oeis.org

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

Offset: 1

Vaclav Kotesovec, Mar 29 2019

			2.07422504479637891390708968594384056977125337962227288334734036988361960596259...

Eric Weisstein's World of Mathematics, Infinite Product.

Cf. A156648, A073017, A258870, A258871.

evalf(Product(1 + 1/j^5, j = 1..infinity), 120);

RealDigits[Chop[N[Product[(1 + 1/n^5), {n, 1, Infinity}], 120]]][[1]]
With[{g = GoldenRatio}, Chop[N[1/(Gamma[1/(2*g^2) - I*5^(1/4)/(2*Sqrt[g])] * Gamma[g^2/2 + I*5^(1/4) * Sqrt[g]/2] * Gamma[g^2/2 - I*5^(1/4) * Sqrt[g]/2] * Gamma[1/(2*g^2) + I*5^(1/4)/(2*Sqrt[g])]), 120]]]
N[1/Abs[Gamma[Exp[2*Pi*I/5]]*Gamma[Exp[6*Pi*I/5]]]^2, 120] (* Vaclav Kotesovec, Apr 27 2020 *)

default(realprecision, 120); exp(sumalt(j=1, -(-1)^j*zeta(5*j)/j))

Equals exp(Sum_{j>=1} (-(-1)^j*Zeta(5*j)/j)).

Equals 1/(Gamma(1/(2*phi^2) - i*(5^(1/4)/(2*sqrt(phi)))) * Gamma(phi^2/2 + i*5^(1/4)*(sqrt(phi)/2)) * Gamma(phi^2/2 - i*5^(1/4)*(sqrt(phi)/2)) * Gamma(1/(2*phi^2) + i*(5^(1/4)/(2*sqrt(phi))))), where i is the imaginary unit and phi = A001622 = (1+sqrt(5))/2 is the golden ratio.

A354054 a(n) = Product_{k=0..n} (k^6 + 1).

Original entry on oeis.org

1, 2, 130, 94900, 388805300, 6075471617800, 283463279271694600, 33349454806314869690000, 8742392830201411514885050000, 4646074730467898538293540742100000, 4646079376542629006192079035640742100000, 8230817672466612927467651920537784356160200000

Offset: 0

Vaclav Kotesovec, May 16 2022

nonn

Winston de Greef, Table of n, a(n) for n = 0..103

Cf. A002604, A101686, A255433, A255434, A258871, A354052.

A354054 := proc(n)
    mul( k^6+1,k=0..n) ;
end proc:
seq(A354054(n),n=0..40) ; # R. J. Mathar, Jul 17 2023

Table[Product[k^6 + 1, {k, 0, n}], {n, 0, 15}]

PARI
```
a(n) = prod(k=1, n, k^6+1);
```

a(n) ~ (2*sinh(2*Pi) - 4*sinh(Pi)*cos(sqrt(3)*Pi)) * n^(6*n + 3) / exp(6*n).

a(n) ~ A258871 * (n!)^6.

A371465 Decimal expansion of Product_{k>=1} (1 - 1/(2*k+1)^6).

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Mar 31 2024

			0.9985530274491360545713131760527821550224491...

Cf. A003881, A175617, A258871, A371219.

RealDigits[(1/24) Pi Cosh[Sqrt[3] Pi/2], 10, 100][[1]]

Equals (1/24) * Pi * cosh(sqrt(3)*Pi/2).

cp's OEIS Frontend

A156648 Decimal expansion of Product_{k>=1} (1 + 1/k^2).

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

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A375843 a(n) = Product_{k=0..n} (k^6 + n).

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A307216 Decimal expansion of Product_{k>=1} (1 + 1/k^5).

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