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

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

Author

Gus Wiseman, Aug 21 2025

Keywords

• nonn
• new

Comments

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.

Examples

			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))
		

Crossrefs

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.

Programs

Mathematica

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

Formula

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