cp's OEIS Frontend

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

%I A078506 #44 May 25 2023 08:37:19
%S A078506 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,
%T A078506 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,
%U A078506 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
%N A078506 Decimal expansion of sum of inverses of unrestricted partition function.
%C A078506 One could just as well sum from n >= 0, giving a value one higher. - _Franklin T. Adams-Watters_, Nov 30 2018
%C A078506 Conjecture: this is a transcendental number.  - _Zhi-Wei Sun_, May 24 2023
%H A078506 Vaclav Kotesovec, <a href="/A078506/b078506.txt">Table of n, a(n) for n = 1..1000</a>
%F A078506 Sum_{n>=1} 1/A000041(n) = 2.510597483886...
%e A078506 2.510597483886293953236834727415465451683531944955147681908...
%t A078506 digits = 100; NSum[1/PartitionsP[n], {n, 1, Infinity}, NSumTerms -> 10000, WorkingPrecision -> digits+1] // RealDigits[#, 10, digits]& // First (* _Jean-François Alcover_, Feb 21 2014 *)
%o A078506 (PARI)
%o A078506 default(realprecision,100);
%o A078506 N=10000;  x='x+O('x^N);
%o A078506 v=Vec(Ser( 1/eta(x) ) );
%o A078506 s=sum(n=2,#v, 1.0/v[n] )
%o A078506 (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 */
%Y A078506 Cf. A000041, A237515, A303662.
%K A078506 cons,nonn
%O A078506 1,1
%A A078506 _Ralf Stephan_, Jan 05 2003
%E A078506 Corrected digits from position 32 on by _Ralf Stephan_, Jan 24 2011
%E A078506 More terms from _Jean-François Alcover_, Feb 21 2014