A132268 Decimal expansion of Product_{k>0} (1-1/12^k).

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

Offset

0,1

Links

G. C. Greubel, Table of n, a(n) for n = 0..1200

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*12^n)) = exp(-Sum_{n>0} A000203(n)/(n*12^n)).

Equals (1/12; 1/12)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 05 2015

From Amiram Eldar, May 09 2023: (Start)

Equals sqrt(2*Pi/log(12)) * exp(log(12)/24 - Pi^2/(6*log(12))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(12))) (McIntosh, 1995).

Equals Sum_{n>=0} (-1)^n/A027880(n). (End)

Example

0.9097262689059948886363620469770...

Mathematica

digits = 106; NProduct[1-1/12^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+3] // N[#, digits+3]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/12, 1/12]] (* G. C. Greubel, Dec 05 2015 *)

Prog

(PARI) prodinf(k=1, (1-1/12^k)) \\ Michel Marcus, Dec 05 2015

Crossrefs

Cf. A027880, A048651, A100220, A132019-A132026, A132034-A132038, A132265-A132267, A132323-A132326, A000203.

Sequence in context: A104756 A154994 A237193 * A252851 A241290 A201298

Adjacent sequences: A132265 A132266 A132267 * A132269 A132270 A132271

Keyword

nonn,cons

Author

Hieronymus Fischer, Aug 20 2007

Status

approved