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

1,4

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

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

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 *)

Prog

(PARI) A168111(n) = sumdiv(n, d, (d<n)*numbpart(d)); \\ Antti Karttunen, Nov 14 2017

Crossrefs

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

Sequence in context: A336647 A336645 A349910 * A309491 A109646 A199783

Adjacent sequences: A168108 A168109 A168110 * A168112 A168113 A168114

Keyword

nonn,easy

Author

Omar E. Pol, Nov 22 2009

Extensions

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

New name from Omar E. Pol, Feb 25 2014

Status

approved