A078506 Decimal expansion of sum of inverses of unrestricted partition function.

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

Offset: 1

Author

Ralf Stephan, Jan 05 2003

Keywords

• cons
• nonn

Comments

One could just as well sum from n >= 0, giving a value one higher. - Franklin T. Adams-Watters, Nov 30 2018

Conjecture: this is a transcendental number. - Zhi-Wei Sun, May 24 2023

Examples

			2.510597483886293953236834727415465451683531944955147681908...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000041, A237515, A303662.

Programs

Mathematica

digits = 100; NSum[1/PartitionsP[n], {n, 1, Infinity}, NSumTerms -> 10000, WorkingPrecision -> digits+1] // RealDigits[#, 10, digits]& // First (* Jean-François Alcover, Feb 21 2014 *)

PARI

default(realprecision,100);
N=10000;  x='x+O('x^N);
v=Vec(Ser( 1/eta(x) ) );
s=sum(n=2,#v, 1.0/v[n] )

PARI

{a(n) = if( n<-1, 0, n++; default( realprecision, n+5); floor( suminf( k=1, 1 / numbpart(k)) * 10^n) % 10)} /* Michael Somos, Feb 05 2011 */

Formula

Sum_{n>=1} 1/A000041(n) = 2.510597483886...

Extensions

Corrected digits from position 32 on by Ralf Stephan, Jan 24 2011

More terms from Jean-François Alcover, Feb 21 2014