# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a116217 Showing 1-1 of 1 %I A116217 #20 Aug 10 2020 09:27:57 %S A116217 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 A116217 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 A116217 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 A116217 Decimal expansion of constant Sum_{i,j,k=1..inf} 1/2^(i*j*k). %C A116217 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 A116217 Eric Weisstein's World of Mathematics, Triple Series. %F A116217 Equals Sum_{n=1..infinity} A007425(n)/2^n . - _R. J. Mathar_, Jan 23 2008 %F A116217 From _Amiram Eldar_, Aug 10 2020: (Start) %F A116217 Equals Sum{k>=1} d(k)/(2^k - 1), where d(k) is the number of divisors of k (A000005). %F A116217 Equals Sum_{i,j=1..oo} 1/(2^(i*j) - 1). (End) %e A116217 2.32478477284047906123521768286139304602095134522547605... %t A116217 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 A116217 (PARI): /* Using sum(n=1..infinity, A007425(n)/2^n ) */ %o A116217 lambert2ser(L)= %o A116217 { %o A116217 local(n, t); %o A116217 n = length(L); %o A116217 t = sum(k=1, length(L), O('x^(n+1))+L[k]*'x^k/(1-'x^k) ); %o A116217 t = Vec(t); %o A116217 return( t ); %o A116217 } %o A116217 N=1000; v=vector(N,n,1); /* roughly 1000 bits precision */ %o A116217 t=lambert2ser(lambert2ser(v)); /* ==[1, 3, 3, 6, 3, 9,...] == A007425 */ %o A116217 default(realprecision,floor(N/3.4)); /* factor approx. log(10)/log(2) */ %o A116217 sum(n=1,#v,1.0*t[n]/2^n) %o A116217 /* == 2.324784772840479061235217682861... */ %Y A116217 Cf. A065442 = Decimal expansion of Erdos-Borwein constant Sum_{k=1..inf} 1/(2^k-1). %Y A116217 Cf. A000005, A007425. %K A116217 cons,nonn %O A116217 1,1 %A A116217 _Alexander Adamchuk_, Apr 09 2007 %E A116217 More terms from _R. J. Mathar_, Jan 23 2008 %E A116217 More terms from _Jean-François Alcover_, Feb 13 2013 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE