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

%I A100222 #28 Feb 16 2025 08:32:55
%S A100222 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,
%T A100222 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,
%U A100222 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
%N A100222 Decimal expansion of Product_{k>=1} (1-1/5^k).
%H A100222 G. C. Greubel, <a href="/A100222/b100222.txt">Table of n, a(n) for n = 0..1200</a>
%H A100222 Richard J. McIntosh, <a href="https://doi.org/10.1112/jlms/51.1.120">Some Asymptotic Formulae for q-Hypergeometric Series</a>, Journal of the London Mathematical Society, Vol. 51, No. 1 (1995), pp. 120-136; <a href="https://citeseerx.ist.psu.edu/pdf/4f03a5e304ec19f8a725774525aecd2a78f4ad81">alternative link</a>.
%H A100222 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>.
%F A100222 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
%F A100222 Equals (1/5; 1/5)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Dec 01 2015
%F A100222 From _Amiram Eldar_, May 09 2023: (Start)
%F A100222 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).
%F A100222 Equals Sum_{n>=0} (-1)^n/A027872(n). (End)
%e A100222 0.76033279587123242010148829629266515947434392887320...
%t A100222 (5^(1/24)*EllipticThetaPrime[1, 0, 1/Sqrt[5]]^(1/3))/2^(1/3)
%t A100222 N[QPochhammer[1/5,1/5]] (* _G. C. Greubel_, Dec 01 2015 *)
%o A100222 (PARI) prodinf(k=1, 1 - 1/(5^k)) \\ _Amiram Eldar_, May 09 2023
%Y A100222 Cf. A027872, A048651, A100220, A100221.
%Y A100222 Cf. A000203, A100220, A100221, A132020, A132034, A132035, A132036, A132037, A132038, A258460.
%K A100222 nonn,cons
%O A100222 0,1
%A A100222 _Eric W. Weisstein_, Nov 09 2004