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

1,2

Comments

The asymptotic density of the occurrences of 0's in this sequence is 1 - Product_{p prime} (1 - 1/p^(A000041(p)+1)) = 0.13580468148150748566... . - Amiram Eldar, Nov 11 2025

Links

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Formula

Multiplicative with a(p^e) = e! * binomial(A000041(p), e). - Andrew Howroyd, Nov 10 2025

Example

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))

Mathematica

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

Prog

(PARI) a(n) = { my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); e!*binomial(numbpart(p), e)) } \\ Andrew Howroyd, Nov 10 2025

Crossrefs

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.

Sequence in context: A260724 A256581 A122697 * A129022 A210564 A208930

Adjacent sequences: A387130 A387131 A387132 * A387134 A387135 A387136

Keyword

nonn,mult

Author

Gus Wiseman, Aug 26 2025

Status

approved