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

1,3

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.

The asymptotic density of the occurrences of 0's in this sequence is 1 - Product_{k>=1} (1 - 1/prime(k)^(A000041(k)+1)) = 0.27893946566270364212... . - Amiram Eldar, Nov 11 2025

Links

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

Formula

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

Example

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))

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}]

Prog

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

Crossrefs

Twice partitions of this type are counted by A296122.

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.

Positions of zero are A387576, counted by A387328.

Positions of nonzero terms are A387577, counted by A387134.

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, A276078, A276079, A299201, A335433, A335448, A355745, A387135.

Sequence in context: A169646 A349125 A231117 * A021496 A241830 A151929

Adjacent sequences: A387107 A387108 A387109 * A387111 A387112 A387113

Keyword

nonn,mult

Author

Gus Wiseman, Aug 18 2025

Status

approved