A322529 -id:A322529 - cp's OEIS frontend

A322526 Number of integer partitions of n whose product of parts is a squarefree number.

1, 1, 2, 3, 3, 5, 6, 8, 9, 10, 13, 15, 17, 21, 24, 27, 30, 36, 41, 46, 51, 57, 65, 73, 82, 90, 101, 109, 121, 134, 150, 164, 177, 193, 214, 232, 253, 278, 300, 324, 351, 386, 419, 452, 484, 521, 563, 610, 658, 706, 758, 809, 868, 938, 1006, 1071, 1140, 1220, 1307

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

The parts of such a partition must also be squarefree and distinct except for any number of 1's.

			The a(8) = 9 partitions are (53), (71), (521), (611), (5111), (32111), (311111), (2111111), (11111111). Missing from this list are (8), (62), (44), (431), (422), (4211), (41111), (332), (3311), (3221), (2222), (22211), (221111).

Partial sums of A322530.

Cf. A001055, A003963, A005117, A064573 , A073576, A096276, A301987, A302505, A319005, A319916, A320322, A322527, A322529, A322531.

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

A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

Original entry on oeis.org

1, 0, 1, 1, 0, 2, 1, 2, 1, 1, 3, 2, 2, 4, 3, 3, 3, 6, 5, 5, 5, 6, 8, 8, 9, 8, 11, 8, 12, 13, 16, 14, 13, 16, 21, 18, 21, 25, 22, 24, 27, 35, 33, 33, 32, 37, 42, 47, 48, 48, 52, 51, 59, 70, 68, 65, 69, 80, 87, 90, 103, 100, 96, 103, 123, 128, 135, 136, 132, 153

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

Such a partition must be strict and its parts must also be squarefree.

			The a(26) = 11 integer partitions:
  (26),
  (15,11), (19,7), (21,5), (23,3),
  (13,7,6), (13,10,3), (13,11,2), (17,7,2), (19,5,2),
  (11,7,5,3).

First differences of A322526.

Cf. A002865, A003963, A005117, A064573, A072774, A073576, A302505, A319005, A319057, A319169, A320322, A322527, A322528, A322529.

Table[Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&SquareFreeQ[Times@@#]&]],{n,30}]

A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 5, 4, 7, 2, 7, 4, 7, 7, 9, 3, 10, 5, 12, 9, 8, 6, 14, 10, 12, 10, 14, 11, 20, 13, 18, 13, 16, 16, 25, 16, 19, 20, 26, 18, 30, 19, 27, 26, 27, 22, 38, 30, 37, 28, 38, 32, 43, 37, 46, 40, 47, 40, 66, 49, 58, 56, 64, 56, 73, 58, 76, 70, 85

Offset: 0

Gus Wiseman, Dec 14 2018

nonn

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (2)   (3)    (4)     (5)      (6)       (7)        (8)
       (11)  (111)  (22)    (32)     (33)      (52)       (44)
                    (1111)  (11111)  (222)     (1111111)  (53)
                                     (111111)             (2222)
                                                          (11111111)

Cf. A002865, A003963, A005117, A064573 , A072774, A295193, A302505, A306017, A319169, A320322, A322526, A322527, A322529, A322530, A322531.

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

More terms from Alois P. Heinz, Dec 14 2018

cp's OEIS Frontend

A322526 Number of integer partitions of n whose product of parts is a squarefree number.

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A322530 Number of integer partitions of n with no 1's whose product of parts is a squarefree number.

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A322528 Number of integer partitions of n whose parts all have the same number of prime factors (counted with multiplicity) and whose product of parts is a power of a squarefree number (A072774).

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Author

Keywords

Examples

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Mathematica

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