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A132025 Decimal expansion of Product_{k>=0} 1-1/(2*9^k). 3

%I #11 May 08 2023 02:27:26

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

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

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

%N Decimal expansion of Product_{k>=0} 1-1/(2*9^k).

%F Equals lim inf_{n->oo} Product_{k=0..floor(log_9(n))} floor(n/9^k)*9^k/n.

%F Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^(1/2*(1+floor(log_9(n)))*floor(log_9(n))).

%F Equals lim inf_{n->oo} A132033(n)/n^(1+floor(log_9(n)))*9^A000217(floor(log_9(n))).

%F Equals (1/2)*exp(-Sum_{n>0} 9^(-n)*Sum_{k|n} 1/(k*2^k)).

%F Equals lim inf_{n->oo} A132033(n)/A132033(n+1).

%F Equals Product_{n>=1} (1 - 1/A270369(n)). - _Amiram Eldar_, May 08 2023

%e 0.4689451783670236932832800...

%t digits = 103; NProduct[1-1/(2*9^k), {k, 0, Infinity}, NProductFactors -> 100, WorkingPrecision -> digits+5] // N[#, digits+5]& // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 18 2014 *)

%t RealDigits[QPochhammer[1/2, 1/9], 10, 120][[1]] (* _Amiram Eldar_, May 08 2023 *)

%Y Cf. A048651, A098844, A067080, A132019, A132026, A132033, A132037, A000217, A270369.

%K nonn,cons

%O 0,1

%A _Hieronymus Fischer_, Aug 14 2007

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Last modified August 28 14:49 EDT 2024. Contains 375507 sequences. (Running on oeis4.)