A132034 Decimal expansion of Product_{k>0} (1-1/6^k).

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

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*6^n) ).

Equals (1/6; 1/6)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 30 2015

From Amiram Eldar, May 09 2023: (Start)

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

Equals Sum_{n>=0} (-1)^n/A027873(n). (End)

Example

0.805687728162164940923750...

Mathematica

digits = 103; NProduct[1-1/6^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+20] // N[#, digits+20]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[1/6, 1/6]] (* G. C. Greubel, Nov 30 2015 *)

Prog

(PARI) prodinf(x=1, 1-(1/6)^x) \\ Altug Alkan, Dec 01 2015

Crossrefs

Cf. A000203, A027873, A048651, A067080, A098844, A100220, A132022, A132026, A132038.

Sequence in context: A146293 A019724 A195400 * A190281 A107950 A336798

Adjacent sequences: A132031 A132032 A132033 * A132035 A132036 A132037

Keyword

nonn,cons

Author

Hieronymus Fischer, Aug 14 2007

Status

approved