A132037 Decimal expansion of Product_{k>0} (1-1/9^k).

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

Offset

0,1

Links

Table of n, a(n) for n=0..102.

Richard J. McIntosh, Some Asymptotic Formulae for q-Hypergeometric Series, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; alternative link.

Formula

Equals exp(-Sum_{n>0} sigma_1(n)/(n*9^n)) = exp(-Sum_{n>0} A000203(n)/(n*9^n)).

Equals Sum_{n>=0} A010815(n)/9^n. - R. J. Mathar, Mar 04 2009

From Amiram Eldar, May 09 2023: (Start)

Equals sqrt(Pi/log(3)) * exp(log(3)/12 - Pi^2/(12*log(3))) * Product_{k>=1} (1 - exp(-2*k*Pi^2/log(3))) (McIntosh, 1995).

Equals Sum_{n>=0} (-1)^n/A027877(n). (End)

Example

0.8765603540359642058360198...

Mathematica

digits = 103; NProduct[1-1/9^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
RealDigits[QPochhammer[1/9], 10, 100][[1]] (* Amiram Eldar, May 09 2023 *)

Prog

(PARI) prodinf(k=1, 1 - 1/(9^k)) \\ Amiram Eldar, May 09 2023

Crossrefs

Cf. A010815, A027877, A048651, A100220, A132025, A132026, A132038, A098844, A067080, A000203.

Sequence in context: A132672 A212410 A328759 * A247095 A124597 A347903

Adjacent sequences: A132034 A132035 A132036 * A132038 A132039 A132040

Keyword

nonn,cons

Author

Hieronymus Fischer, Aug 14 2007

Status

approved