A175616 -id:A175616

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A334411

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

+10
7

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

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)

LINKS

Table of n, a(n) for n=1..105.

FORMULA

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).

EXAMPLE

2.00815605499274531514903948232341369211953215983095097877074299617422...

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

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

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Apr 27 2020

STATUS

approved

A339745

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

+10
1

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..105.

FORMULA

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).

EXAMPLE

0.99900544248098947527378453585422724586059097385364736908229...

MAPLE

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);

MATHEMATICA

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]]

PROG

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

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Vaclav Kotesovec, Dec 15 2020

STATUS

approved

A346745

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

+10
0

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

(list; constant; graph; refs; listen; history; text; internal format)

OFFSET

0,1

LINKS

Table of n, a(n) for n=0..86.

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

FORMULA

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

EXAMPLE

0.999753913921893256003448570641909727180...

MAPLE

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

MATHEMATICA

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 *)

PROG

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

CROSSREFS

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

KEYWORD

nonn,cons

AUTHOR

Sean A. Irvine, Jul 31 2021

STATUS

approved

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