A307216 -id:A307216 - cp's OEIS frontend

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

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).

A375842 a(n) = Product_{k=0..n} (k^5 + n).

Original entry on oeis.org

0, 2, 204, 103320, 182819520, 886613565600, 9973411835945040, 230723130088707984000, 10026790739932043582668800, 762878670971305314645055065600, 96049580303627434572710125880376000, 19110279123993980912852049573590159155200

Offset: 0

Vaclav Kotesovec, Aug 31 2024

nonn

Cf. A307216, A255435.

Cf. A375839, A375840, A375841, A375843, A375844, A375845.

Table[Product[k^5 + n, {k, 0, n}], {n, 0, 15}]

a(n) ~ n^(5*n + 3) / exp(5*n - Pi * sqrt(2*(1+sqrt(5))) * n^(1/5) / 5^(1/4)).

cp's OEIS Frontend

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

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