A387326 -id:A387326 - cp's OEIS frontend

A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

1, 2, 3, 2, 7, 6, 15, 0, 6, 14, 56, 6, 101, 30, 21, 0, 297, 12, 490, 14, 45, 112, 1255, 0, 42, 202, 6, 30, 4565, 42, 6842, 0, 168, 594, 105, 12, 21637, 980, 303, 0, 44583, 90, 63261, 112, 42, 2510, 124754, 0, 210, 84, 891, 202, 329931, 12, 392, 0, 1470, 9130

Offset: 1

Gus Wiseman, Aug 26 2025

			The prime factors of 9 are (3,3), and the a(9) = 6 choices are:
  ((3),(2,1))
  ((3),(1,1,1))
  ((2,1),(3))
  ((2,1),(1,1,1))
  ((1,1,1),(3))
  ((1,1,1),(2,1))

For prime factors instead of partitions we have A008966, see A355741.
Twice partitions of this type are counted by A296122.
For prime indices instead of factors we have A387110, see A387136.
For strict partitions and prime indices we have A387115.
For constant partitions and prime indices we have A387120.
Positions of zero are A387326, for indices apparently A276079 (complement A276078).
Allowing repeated partitions gives A387327, see A299200, A357977.
A000041 counts integer partitions, strict A000009.
A003963 multiplies together prime indices.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A120383 lists numbers divisible by all of their prime indices.
A289509 lists numbers with relatively prime prime indices.
Cf. A063834, A261049, A299201, A335433, A335448, A355739, A383706, A387111, A387135.

Table[Length[Select[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]],UnsameQ@@#&]],{n,30}]

A387327 Number of ways to choose an integer partition of each prime factor of n (with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 7, 6, 15, 8, 9, 14, 56, 12, 101, 30, 21, 16, 297, 18, 490, 28, 45, 112, 1255, 24, 49, 202, 27, 60, 4565, 42, 6842, 32, 168, 594, 105, 36, 21637, 980, 303, 56, 44583, 90, 63261, 224, 63, 2510, 124754, 48, 225, 98, 891, 404, 329931, 54, 392, 120

Offset: 1

Gus Wiseman, Sep 05 2025

			The a(1) = 1 through a(7) = 15 ways:
  (1)  (2)   (3)    (2)(2)    (5)      (2)(3)     (7)
       (11)  (21)   (11)(2)   (32)     (11)(3)    (43)
             (111)  (2)(11)   (41)     (2)(21)    (52)
                    (11)(11)  (221)    (11)(21)   (61)
                              (311)    (2)(111)   (322)
                              (2111)   (11)(111)  (331)
                              (11111)             (421)
                                                  (511)
                                                  (2221)
                                                  (3211)
                                                  (4111)
                                                  (22111)
                                                  (31111)
                                                  (211111)
                                                  (1111111)

For constant partitions we have A061142, for prime indices A355731.
For prime indices instead of factors we have A299200.
The version for distinct choices is A387133, zeros A387326.
A000041 counts integer partitions, strict A000009.
A112798 lists prime indices, row sums A056239 or A066328, lengths A001222.
A387110 counts choices of distinct distinct integer partitions of each prime index.
Cf. A063834, A120383, A299201, A335433, A335448, A355741, A357977, A383706, A387115, A387120, A387136.

Table[Length[Tuples[IntegerPartitions/@Flatten[ConstantArray@@@FactorInteger[n]]]],{n,30}]

cp's OEIS Frontend

A387133 Number of ways to choose a sequence of distinct integer partitions, one of each prime factor of n (with multiplicity).

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