A144300 Number of partitions of n minus number of divisors of n.

0, 0, 1, 2, 5, 7, 13, 18, 27, 38, 54, 71, 99, 131, 172, 226, 295, 379, 488, 621, 788, 998, 1253, 1567, 1955, 2432, 3006, 3712, 4563, 5596, 6840, 8343, 10139, 12306, 14879, 17968, 21635, 26011, 31181, 37330, 44581, 53166, 63259, 75169, 89128, 105554, 124752

Offset: 1

Author

Omar E. Pol, Sep 17 2008

Keywords

• easy
• nonn

Comments

a(n) is also the number of partitions of n with at least one distinct part (i.e., not all parts are equal).

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..1000
• Omar E. Pol, The shell model of partitions

Crossrefs

Cf. A000005, A000041, A135010, A138121, A195364.

A182114(n,n-1) = a(n). - Alois P. Heinz, Nov 02 2012

Programs

Maple

with(numtheory): b:= proc(n) option remember; `if`(n=0, 1, add(add(d, d=divisors(j)) *b(n-j), j=1..n)/n) end: a:= n-> b(n)- tau(n):
seq(a(n), n=1..50);  # Alois P. Heinz, Oct 07 2008

Mathematica

Table[PartitionsP[n]-DivisorSigma[0,n],{n,50}] (* Harvey P. Dale, Apr 10 2014 *)

PARI

al(n)=vector(n,k,numbpart(k)-numdiv(k))

Python

from sympy import npartitions, divisor_count
def A144300(n): return npartitions(n)-divisor_count(n) # Chai Wah Wu, Oct 16 2023

Formula

a(n) = p(n) - d(n) = A000041(n) - A000005(n).