A132324 Decimal expansion of Product_{k>=1} (1+1/3^k).

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

Offset: 1

Author

Hieronymus Fischer, Aug 20 2007

Keywords

• nonn
• cons

Comments

Half the constant A132323.

Examples

			1.56493401856701153793884910...
		

Links

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

Crossrefs

Cf. A027871, A079555, A100220, A132019-A132026, A132034-A132038, A132265-A132268, A132323-A132326, A132327, A132328, A000593.

Programs

Mathematica

digits = 105; NProduct[1+1/3^k, {k, 1, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 18 2014 *)
N[QPochhammer[-1/3,1/3]] (* G. C. Greubel, Dec 01 2015 *)

Formula

(1/2)*lim sup Product{k=0..floor(log_3(n))} (1+1/floor(n/3^k)) for n-->oo.

(1/2)*lim sup A132327(n)/A132027(n) for n-->oo.

(1/2)*lim sup A132327(n)/n^((1+log_3(n))/2) for n-->oo.

(1/2)*lim sup A132328(n)/n^((log_3(n)-1)/2) for n-->oo.

exp(Sum_{n>0} 3^(-n)*Sum_{k|n} -(-1)^k/k) = exp(Sum_{n>0} A000593(n)/(n*3^n)).

(1/2)*lim sup A132327(n+1)/A132327(n) = 1.56493401856701153793884910... for n-->oo.

Equals (-1/3; 1/3){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 01 2015

From Amiram Eldar, Feb 19 2022: (Start)

Equals (sqrt(2)/2) * exp(log(3)/24 + Pi^2/(12*log(3))) * Product_{k>=1} (1 - exp(-2*(2*k-1)*Pi^2/log(3))) (McIntosh, 1995).

Equals Sum_{n>=0} 1/A027871(n). (End)