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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.
4
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
0,3
COMMENTS
Such a partition must be strict (unless it is all 1's) and its parts must also be squarefree.
LINKS
Lucas A. Brown, Python program.
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}]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Dec 14 2018
EXTENSIONS
a(51)-a(69) from Jinyuan Wang, Jun 27 2020
a(70) onwards from Lucas A. Brown, Aug 17 2024
STATUS
approved