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%I #20 Aug 10 2020 09:27:57
%S 2,3,2,4,7,8,4,7,7,2,8,4,0,4,7,9,0,6,1,2,3,5,2,1,7,6,8,2,8,6,1,3,9,3,
%T 0,4,6,0,2,0,9,5,1,3,4,5,2,2,5,4,7,6,0,5,3,6,0,1,4,6,9,4,6,4,4,4,1,9,
%U 2,2,0,2,0,0,4,6,3,9,7,7,0,3,1,7,3,6,9,8,8,4,0,1,5,1,2,7,7,2,8,2,6,8,8,3,1
%N Decimal expansion of constant Sum_{i,j,k=1..inf} 1/2^(i*j*k).
%C This constant is a sum of triple series Sum[Sum[Sum[1/2^(i*j*k),{i,1,Infinity}],{j,1,Infinity}],{k,1,Infinity}] = 2.3247847... It is similar to Erdos-Borwein constant Sum[Sum[1/2^(i*j),{i,1,Infinity}],{j,1,Infinity}] = Sum[1/(2^k-1),{k,1,Infinity}] = 1.60669515...
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/TripleSeries.html">Triple Series</a>.
%F Equals Sum_{n=1..infinity} A007425(n)/2^n . - _R. J. Mathar_, Jan 23 2008
%F From _Amiram Eldar_, Aug 10 2020: (Start)
%F Equals Sum{k>=1} d(k)/(2^k - 1), where d(k) is the number of divisors of k (A000005).
%F Equals Sum_{i,j=1..oo} 1/(2^(i*j) - 1). (End)
%e 2.32478477284047906123521768286139304602095134522547605...
%t digits = 105; Clear[s]; s[n_] := s[n] = 2*NSum[1/(2^(j*k) - 1), {j, 1, n}, {k, 1, j-1}, WorkingPrecision -> digits+10, NSumTerms -> 100] + NSum[1/(2^j^2 - 1), {j, 1, n}, WorkingPrecision -> digits+10, NSumTerms -> 100] // RealDigits[#, 10, digits]& // First; s[n=100]; While[s[n] != s[n-100], n = n+100]; s[n] (* _Jean-François Alcover_, Feb 13 2013 *)
%o (PARI): /* Using sum(n=1..infinity, A007425(n)/2^n ) */
%o lambert2ser(L)=
%o {
%o local(n, t);
%o n = length(L);
%o t = sum(k=1, length(L), O('x^(n+1))+L[k]*'x^k/(1-'x^k) );
%o t = Vec(t);
%o return( t );
%o }
%o N=1000; v=vector(N,n,1); /* roughly 1000 bits precision */
%o t=lambert2ser(lambert2ser(v)); /* ==[1, 3, 3, 6, 3, 9,...] == A007425 */
%o default(realprecision,floor(N/3.4)); /* factor approx. log(10)/log(2) */
%o sum(n=1,#v,1.0*t[n]/2^n)
%o /* == 2.324784772840479061235217682861... */
%Y Cf. A065442 = Decimal expansion of Erdos-Borwein constant Sum_{k=1..inf} 1/(2^k-1).
%Y Cf. A000005, A007425.
%K cons,nonn
%O 1,1
%A _Alexander Adamchuk_, Apr 09 2007
%E More terms from _R. J. Mathar_, Jan 23 2008
%E More terms from _Jean-François Alcover_, Feb 13 2013
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