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 A048651 Decimal expansion of Product_{k >= 1} (1 - 1/2^k). 73

%I

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

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

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

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

%C This is the limiting probability that a large random binary matrix is nonsingular (cf. A002884).

%D S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 354-361.

%H Harry J. Smith, <a href="/A048651/b048651.txt">Table of n, a(n) for n = 0..20000</a>

%H S. R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/dig/dig.html">Digital Search Tree Constants</a> [Broken link]

%H Victor S. Miller, <a href="http://arxiv.org/abs/1606.09299">Counting Matrices that are Squares</a>, arXiv:1606.09299 [math.GR], 2016.

%H Kent E. Morrison, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Morrison/morrison37.html">Integer Sequences and Matrices Over Finite Fields</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.1.

%H László Tóth, <a href="https://arxiv.org/abs/1608.00795">Alternating sums concerning multiplicative arithmetic functions</a>, arXiv preprint arXiv:1608.00795 [math.NT], 2016.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TreeSearching.html">Tree Searching</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/InfiniteProduct.html">Infinite Product</a>

%F exp(-Sum_{k>0} sigma_1(k)/k*2^(-k)) = exp(-Sum_{k>0} A000203(k)/k*2^(-k)). - _Hieronymus Fischer_, Jul 28 2007

%F Lim inf Product_{k=0..floor(log_2(n))} floor(n/2^k)*2^k/n for n->oo. - _Hieronymus Fischer_, Aug 13 2007

%F Lim inf A098844(n)/n^(1+floor(log_2(n)))*2^(1/2*(1+floor(log_2(n)))*floor(log_2(n))) for n->oo. - _Hieronymus Fischer_, Aug 13 2007

%F Lim inf A098844(n)/n^(1+floor(log_2(n)))*2^A000217(floor(log_2(n)) for n->oo. - _Hieronymus Fischer_, Aug 13 2007

%F Lim inf A098844(n)/(n+1)^((1+log_2(n+1))/2) for n->oo. - _Hieronymus Fischer_, Aug 13 2007

%F (1/2)*exp(-Sum_{n>0} 2^(-n)*Sum_{k|n} 1/(k*2^k)). - _Hieronymus Fischer_, Aug 13 2007

%F Limit of A177510(n)/A000079(n-1) as n->infinity (conjecture). - _Mats Granvik_, Mar 27 2011

%F Product_{k >= 1} (1-1/2^k) = (1/2; 1/2)_{infinity}, where (a;q)_{infinity} is the q-Pochhammer symbol. - _G. C. Greubel_, Nov 27 2015

%F exp(Sum_{n>=1}(1/n/(1 - 2^n))) (according to Mathematica). - _Mats Granvik_, Sep 07 2016

%F (Sum_{k>0} (4^k-1)/(Product_{i=1..k} ((4^i-1)*(2*4^i-1))))*2 = 2/7 + 2/(3*7*31) + 2/(3*7*15*31*127)+2/(3*7*15*31*63*127*511) + ... (conjecture). - _Werner Schulte_, Dec 22 2016

%e (1/2) (3/4) (7/8) (15/16) ... = 0.288788095086602421278899721929230780088911904840685784114741...

%t RealDigits[ Product[1 - 1/2^i, {i, 100}], 10, 111][[1]] (* _Robert G. Wilson v_, May 25 2011 *)

%t RealDigits[QPochhammer[1/2], 10, 100][[1]] (* _Jean-François Alcover_, Nov 18 2015 *)

%o (PARI) { default(realprecision, 20080); x=prodinf(k=1, -1/2^k, 1); x*=10; for (n=0, 20000, d=floor(x); x=(x-d)*10; write("b048651.txt", n, " ", d)); } \\ _Harry J. Smith_, May 07 2009

%Y Cf. A002884, A005329, A048652, A098844, A067080, A100220, A132019, A132020, A132026, A132038, A070933, A261584.

%K nonn,cons

%O 0,1

%A _N. J. A. Sloane_

%E Corrected by _Hieronymus Fischer_, Jul 28 2007

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