A387115 Number of ways to choose a sequence of distinct strict integer partitions, one of each prime index of n.

1, 1, 1, 0, 2, 1, 2, 0, 0, 2, 3, 0, 4, 2, 2, 0, 5, 0, 6, 0, 2, 3, 8, 0, 2, 4, 0, 0, 10, 2, 12, 0, 3, 5, 4, 0, 15, 6, 4, 0, 18, 2, 22, 0, 0, 8, 27, 0, 2, 2, 5, 0, 32, 0, 6, 0, 6, 10, 38, 0, 46, 12, 0, 0, 8, 3, 54, 0, 8, 4, 64, 0, 76, 15, 2, 0, 6, 4, 89, 0, 0

Offset

1,5

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)^(A000009(k)+1)) = 0.34064159435571437619... . - 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(A000009(k), e). - Andrew Howroyd, Nov 10 2025

Example

The prime indices of 15 are (2,3), and there are a(15) = 2 choices:
  ((2),(3))
  ((2),(2,1))
The prime indices of 121 are (5,5), and there are a(121) = 6 choices:
  ((5),(4,1))
  ((5),(3,2))
  ((4,1),(5))
  ((4,1),(3,2))
  ((3,2),(5))
  ((3,2),(4,1))

Mathematica

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

Prog

(PARI) \\ here b(n) = A000009(n).
b(n)=my(A=O(x*x^n)); polcoef(eta(x^2 + A)/eta(x + A), n)
a(n) = { my(f=factor(n)); prod(i=1, #f~, my([p, e]=f[i, ]); e!*binomial(b(primepi(p)), e)) } \\ Andrew Howroyd, Nov 10 2025

Crossrefs

For divisors instead of partitions we have A355739, see A355740, A355733, A355734.

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

Twice-partitions of this type are counted by A358914, strict case of A270995.

The disjoint case is A383706.

Allowing non-strict partitions gives A387110, for prime factors A387133.

For initial intervals instead of strict partitions we have A387111.

For constant instead of strict partitions we have A387120.

Positions of 0 are A387176 (non-choosable), complement A387177 (choosable).

A000041 counts integer partitions, strict A000009.

A003963 multiplies together the prime indices of n.

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. A000720, A063834, A261049, A335433, A335448, A355731, A355741, A355742, A355744, A357980.

Sequence in context: A029391 A058087 A240753 * A202149 A236533 A298187

Adjacent sequences: A387112 A387113 A387114 * A387116 A387117 A387118

Keyword

nonn,easy,mult

Author

Gus Wiseman, Aug 20 2025

Status

approved