A100220 Decimal expansion of Product_{k>=1} (1 - 1/3^k).
5, 6, 0, 1, 2, 6, 0, 7, 7, 9, 2, 7, 9, 4, 8, 9, 4, 4, 9, 6, 9, 7, 9, 2, 2, 4, 3, 3, 1, 4, 1, 4, 0, 0, 1, 4, 3, 7, 9, 7, 3, 6, 3, 3, 3, 7, 9, 8, 3, 6, 2, 4, 6, 4, 4, 6, 2, 9, 5, 6, 2, 9, 7, 3, 1, 7, 5, 3, 3, 9, 6, 3, 0, 8, 9, 0, 3, 3, 7, 9, 4, 7, 0, 7, 7, 1, 6, 9, 1, 8, 7, 7, 0, 5, 3, 6, 7, 4, 3, 3, 4, 8
Offset: 0
Examples
0.56012607792794894496979224331414001437973633379836...
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.
- Kent E. Morrison, Integer Sequences and Matrices Over Finite Fields, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.
- Eric Weisstein's World of Mathematics, Infinite Product.
Crossrefs
Programs
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Mathematica
N[(3^(1/24)*EllipticThetaPrime[1, 0, 1/Sqrt[3]]^(1/3))/2^(1/3)] N[QPochhammer[1/3,1/3]] (* G. C. Greubel, Nov 27 2015 *)
Formula
exp(-Sum_{k > 0} sigma_1(k)/k/3^k) = exp(-Sum_{k > 0} A000203(k)/k/3^k). - Hieronymus Fischer, Aug 07 2007
Product_{k >= 1} (1 - 1/3^k) = (1/3; 1/3){infinity}, where (a;q){infinity} is the q-Pochhammer symbol. - G. C. Greubel, Nov 27 2015
From Peter Bala, Jan 18 2021: (Start)
Constant C = (1 - 1/3)*Sum_{n >= 0} (-1/3)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*Sum_{n >= 0} (-1/9)^n/Product_{k = 1..n} (3^k - 1);
C = (1 - 1/3)*(1 - 1/9)*(1 - 1/27)*Sum_{n >= 0} (-1/27)^n/Product_{k = 1..n} (3^k - 1), and so on. (End)
From Amiram Eldar, Feb 19 2022: (Start)
Equals sqrt(2*Pi/log(3)) * exp(log(3)/24 - Pi^2/(6*log(3))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(3))) (McIntosh, 1995).
Equals Sum_{n>=0} (-1)^n/A027871(n). (End)
Comments