A322529 Number of integer partitions of n whose parts all have the same number of prime factors (counted with or without multiplicity) and whose product of parts is a squarefree number.

1, 1, 2, 2, 1, 3, 2, 3, 2, 2, 4, 2, 3, 3, 4, 4, 4, 3, 5, 4, 5, 6, 6, 6, 6, 6, 8, 6, 7, 9, 8, 11, 8, 11, 11, 11, 12, 13, 13, 15, 13, 17, 17, 18, 18, 17, 20, 22, 21, 24, 24, 24, 26, 29, 28, 33, 30, 35, 34, 38, 38, 45, 42, 43, 45, 48, 52, 54, 55, 59, 59, 65, 65, 72, 73

Offset: 0

Author

Gus Wiseman, Dec 14 2018

Keywords

• nonn

Comments

Such a partition must be strict (unless it is all 1's) and its parts must also be squarefree.

Examples

			The a(30) = 8 integer partitions:
  (30),
  (17,13),(19,11),(23,7),
  (17,11,2),(23,5,2),
  (13,7,5,3,2),
  (1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1).
		

Links

• Lucas A. Brown, Table of n, a(n) for n = 0..133
• Lucas A. Brown, Python program.

Crossrefs

Cf. A002865, A003963, A005117, A038041, A064573, A073576, A302505, A319056, A320322, A321717, A321718, A322526, A322527, A322528, A322530, A322531.

Programs

Mathematica

Table[Length[Select[IntegerPartitions[n],And[SameQ@@PrimeOmega/@#,SquareFreeQ[Times@@#]]&]],{n,30}]

Extensions

a(51)-a(69) from Jinyuan Wang, Jun 27 2020

a(70) onwards from Lucas A. Brown, Aug 17 2024