A387110 -id:A387110 - cp's OEIS frontend

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

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}]

A387116 Number of ways to choose a constant sequence of integer partitions, one of each prime index of n.

Original entry on oeis.org

1, 1, 2, 1, 3, 0, 5, 1, 2, 0, 7, 0, 11, 0, 0, 1, 15, 0, 22, 0, 0, 0, 30, 0, 3, 0, 2, 0, 42, 0, 56, 1, 0, 0, 0, 0, 77, 0, 0, 0, 101, 0, 135, 0, 0, 0, 176, 0, 5, 0, 0, 0, 231, 0, 0, 0, 0, 0, 297, 0, 385, 0, 0, 1, 0, 0, 490, 0, 0, 0, 627, 0, 792, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Aug 21 2025

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

If n is a prime power prime(x)^y, then a(n) is the number of integer partitions of x; otherwise, a(n) = 0.

			The a(49) = 5 choices:
  ((4),(4))
  ((3,1),(3,1))
  ((2,2),(2,2))
  ((2,1,1),(2,1,1))
  ((1,1,1,1),(1,1,1,1))

Positions of zeros are A024619, complement A000961.
Twice-partitions of this type are counted by A047968, see also A296122.
For initial intervals instead of partitions we have A055396, see also A387111.
This is the constant case of A299200, see also A357977, A357982.
For disjoint instead of constant we have A383706.
For distinct instead of constant we have A387110.
For divisors instead of partitions we have A387114, see also A355731, A355739.
For strict partitions instead of partitions we have A387117.
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. A008284, A063834, A276078, A335433, A335448, A355529, A355741, A387115, A387120, A387133, A387136.

Table[Switch[n,1,1,?PrimePowerQ,PartitionsP[PrimePi[FactorInteger[n][[1,1]]]],,0],{n,100}]

a(n) = A000041(A297109(n)).

cp's OEIS Frontend

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

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