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

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

1,1

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..25000

Simon Plouffe, product(1+1/n**3,n=1..infinity).

S. Ramanujan, Notebook entry.

Formula

Equals cosh(sqrt(3) * Pi / 2)/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

Example

2.42818979209887032873604143617914635811836294478339049763...

Mathematica

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

Prog

(Magma)
m:= 510; R:=RealField(m); SetDefaultRealField(R);
Prune(Reverse(IntegerToSequence(Floor(( Cosh(Sqrt(3)*Pi(R)/2)/Pi(R) )*10^(Floor(m/2)) )))); // G. C. Greubel, Sep 09 2025
(SageMath)
numerical_approx( cosh(sqrt(3)*pi/2)/pi , digits=265 ) # G. C. Greubel, Sep 09 2025

Crossrefs

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

Sequence in context: A354766 A207622 A335573 * A296092 A259111 A374604

Adjacent sequences: A073014 A073015 A073016 * A073018 A073019 A073020

Keyword

cons,nonn

Author

Robert G. Wilson v, Aug 03 2002

Status

approved