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

1,1

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

Links

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

Formula

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

Example

2.510597483886293953236834727415465451683531944955147681908...

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 *)

Prog

(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 */

Crossrefs

Cf. A000041, A237515, A303662.

Sequence in context: A329251 A062627 A011217 * A323909 A124779 A284750

Adjacent sequences: A078503 A078504 A078505 * A078507 A078508 A078509

Keyword

cons,nonn

Author

Ralf Stephan, Jan 05 2003

Extensions

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

More terms from Jean-François Alcover, Feb 21 2014

Status

approved