A116217 Decimal expansion of constant Sum_{i,j,k=1..inf} 1/2^(i*j*k).

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, 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, 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

Offset: 1

Author

Alexander Adamchuk, Apr 09 2007

Keywords

• cons
• nonn

Comments

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...

Examples

			2.32478477284047906123521768286139304602095134522547605...
		

Links

• Eric Weisstein's World of Mathematics, Triple Series.

Crossrefs

Cf. A065442 = Decimal expansion of Erdos-Borwein constant Sum_{k=1..inf} 1/(2^k-1).

Cf. A000005, A007425.

Programs

Mathematica

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 *)

PARI

/* Using sum(n=1..infinity, A007425(n)/2^n )  */
lambert2ser(L)=
{
    local(n, t);
    n = length(L);
    t = sum(k=1, length(L), O('x^(n+1))+L[k]*'x^k/(1-'x^k) );
    t = Vec(t);
    return( t );
}
N=1000; v=vector(N,n,1); /* roughly 1000 bits precision */
t=lambert2ser(lambert2ser(v)); /* ==[1, 3, 3, 6, 3, 9,...] == A007425 */
default(realprecision,floor(N/3.4)); /* factor approx. log(10)/log(2) */
sum(n=1,#v,1.0*t[n]/2^n)
/* == 2.324784772840479061235217682861... */

Formula

Equals Sum_{n=1..infinity} A007425(n)/2^n . - R. J. Mathar, Jan 23 2008

From Amiram Eldar, Aug 10 2020: (Start)

Equals Sum{k>=1} d(k)/(2^k - 1), where d(k) is the number of divisors of k (A000005).

Equals Sum_{i,j=1..oo} 1/(2^(i*j) - 1). (End)

Extensions

More terms from R. J. Mathar, Jan 23 2008

More terms from Jean-François Alcover, Feb 13 2013