A168111 Sum of the partition numbers of the proper divisors of n, with a(1) = 0.

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, 158, 1, 242, 1, 261, 60, 300, 23, 514, 1, 493, 105, 706, 1, 959, 1, 1066, 217, 1258, 1, 1927, 16, 2010, 301, 2545, 1, 3442, 64, 3898, 494, 4568, 1, 6555, 1, 6845, 841, 8610

Offset: 1

Author

Omar E. Pol, Nov 22 2009

Keywords

• nonn
• easy

Comments

Row sums of triangle A168021 except the first column.

Row sums of triangle A168016 except the last column.

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

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

Programs

Maple

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

Mathematica

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

PARI

A168111(n) = sumdiv(n,d,(dAntti Karttunen, Nov 14 2017

Formula

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

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

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

Extensions

Terms beyond a(12) from R. J. Mathar, Jan 25 2010

New name from Omar E. Pol, Feb 25 2014