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

Offset

0,3

Comments

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

Links

Table of n, a(n) for n=0..69.

Example

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).

Mathematica

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

Crossrefs

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

Sequence in context: A128180 A209279 A074754 * A349526 A329949 A333867

Adjacent sequences: A322526 A322527 A322528 * A322530 A322531 A322532

Keyword

nonn,more

Author

Gus Wiseman, Dec 14 2018

Extensions

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

Status

approved