A100222 Decimal expansion of Product_{k>=1} (1-1/5^k).

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

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.

Eric Weisstein's World of Mathematics, Infinite Product.

Formula

Equals exp(-Sum_{k>0} sigma_1(k)/(k*5^k)) = exp(-Sum_{k>0} A000203(k)/(k*5^k)). - Hieronymus Fischer, Aug 07 2007

Equals (1/5; 1/5)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - G. C. Greubel, Dec 01 2015

From Amiram Eldar, May 09 2023: (Start)

Equals sqrt(2*Pi/log(5)) * exp(log(5)/24 - Pi^2/(6*log(5))) * Product_{k>=1} (1 - exp(-4*k*Pi^2/log(5))) (McIntosh, 1995).

Equals Sum_{n>=0} (-1)^n/A027872(n). (End)

Example

0.76033279587123242010148829629266515947434392887320...

Mathematica

(5^(1/24)*EllipticThetaPrime[1, 0, 1/Sqrt[5]]^(1/3))/2^(1/3)
N[QPochhammer[1/5, 1/5]] (* G. C. Greubel, Dec 01 2015 *)

Prog

(PARI) prodinf(k=1, 1 - 1/(5^k)) \\ Amiram Eldar, May 09 2023

Crossrefs

Cf. A027872, A048651, A100220, A100221.

Cf. A000203, A100220, A100221, A132020, A132034, A132035, A132036, A132037, A132038, A258460.

Sequence in context: A003299 A198677 A154017 * A196530 A154850 A011236

Adjacent sequences: A100219 A100220 A100221 * A100223 A100224 A100225

Keyword

nonn,cons

Author

Eric W. Weisstein, Nov 09 2004

Status

approved