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

1,4

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

Formula

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

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 *)

Prog

(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

Crossrefs

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

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

Sequence in context: A176983 A160676 A169690 * A258430 A333242 A045353

Adjacent sequences: A144297 A144298 A144299 * A144301 A144302 A144303

Keyword

easy,nonn

Author

Omar E. Pol, Sep 17 2008

Status

approved