A387110 Number of ways to choose a sequence of distinct integer partitions, one of each prime index of n.

1, 1, 2, 0, 3, 2, 5, 0, 2, 3, 7, 0, 11, 5, 6, 0, 15, 2, 22, 0, 10, 7, 30, 0, 6, 11, 0, 0, 42, 6, 56, 0, 14, 15, 15, 0, 77, 22, 22, 0, 101, 10, 135, 0, 6, 30, 176, 0, 20, 6, 30, 0, 231, 0, 21, 0, 44, 42, 297, 0, 385, 56, 10, 0, 33, 14, 490, 0, 60, 15, 627, 0

Offset: 1

Author

Gus Wiseman, Aug 18 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.

The axiom of choice says that, given any sequence of nonempty sets, it is possible to choose a sequence containing an element from each. In the strict version, the elements of this sequence must be distinct, meaning none is chosen more than once.

Examples

			The prime indices of 9 are (2,2), and there are a(9) = 2 choices:
  ((2),(1,1))
  ((1,1),(2))
The prime indices of 15 are (2,3), and there are a(15) = 5 choices:
  ((2),(3))
  ((2),(2,1))
  ((2),(1,1,1))
  ((1,1),(2,1))
  ((1,1),(1,1,1))
		

Crossrefs

Positions of zeros are A276078 (choosable), complement A276079 (non-choosable).

Allowing repeated partitions gives A299200, A357977, A357982, A357978.

For multiset systems see A355529, A355744, A367771, set systems A367901-A367905.

For divisors see A355731, A355733, A355734, A355735, A355737, A355739, A355740.

For prime factors instead of partitions see A355741, A355742, A387136.

The disjoint case is A383706.

For initial intervals instead of partitions we have A387111.

The case of strict partitions is A387115.

The case of constant partitions is A387120.

Taking each prime factor (instead of index) gives A387133.

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, A261049, A296122, A299201, A335433, A335448, A355745, A387135.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Tuples[IntegerPartitions/@prix[n]],UnsameQ@@#&]],{n,100}]