A168111 - OEIS

%I #28 Oct 08 2024 22:19:09

%S 0,1,1,3,1,6,1,8,4,10,1,22,1,18,11,30,1,47,1,57,19,59,1,121,8,104,34,

%T 158,1,242,1,261,60,300,23,514,1,493,105,706,1,959,1,1066,217,1258,1,

%U 1927,16,2010,301,2545,1,3442,64,3898,494,4568,1,6555,1,6845,841,8610

%N Sum of the partition numbers of the proper divisors of n, with a(1) = 0.

%C Row sums of triangle A168021 except the first column.

%C Row sums of triangle A168016 except the last column.

%H Antti Karttunen, <a href="/A168111/b168111.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = A047968(n) - A000041(n).

%F G.f.: Sum_{n > 0} A000041(n)*x^(2*n)/(1-x^n). - _Mircea Merca_, Feb 24 2014

%F G.f.: x^2 + x^3 + 3*x^4 + x^5 + 6*x^6 + x^7 + 8*x^8 + 4*x^9 + 10*x^10 + x^11 + ... - _Michael Somos_, Feb 24 2014

%p A047968 := proc(n) add(combinat[numbpart](d), d= numtheory[divisors](n) ) ; end proc: A000041 := proc(n) combinat[numbpart](n) ; end proc: A168111 := proc(n) A047968(n)-A000041(n) ; end proc: seq(A168111(n),n=1..90) ; # _R. J. Mathar_, Jan 25 2010

%t a[ n_] := If[n < 1, 0, Sum[ PartitionsP[ d] Boole[ d < n], {d, Divisors @ n}]]; (* _Michael Somos_, Feb 24 2014 *)

%o (PARI) A168111(n) = sumdiv(n,d,(d<n)*numbpart(d)); \\ _Antti Karttunen_, Nov 14 2017

%Y Cf. A000041, A001065, A047968, A168016, A168017, A168018, A168020, A168021.

%K nonn,easy

%O 1,4

%A _Omar E. Pol_, Nov 22 2009

%E Terms beyond a(12) from _R. J. Mathar_, Jan 25 2010

%E New name from _Omar E. Pol_, Feb 25 2014