A387120 Number of ways to choose a different constant integer partition of each prime index of n.

1, 1, 2, 0, 2, 2, 3, 0, 2, 2, 2, 0, 4, 3, 4, 0, 2, 2, 4, 0, 6, 2, 3, 0, 2, 4, 0, 0, 4, 4, 2, 0, 4, 2, 6, 0, 6, 4, 8, 0, 2, 6, 4, 0, 4, 3, 4, 0, 6, 2, 4, 0, 5, 0, 4, 0, 8, 4, 2, 0, 6, 2, 6, 0, 8, 4, 2, 0, 6, 6, 6, 0, 4, 6, 4, 0, 6, 8, 4, 0, 0, 2, 2, 0, 4, 4, 8

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 asymptotic density of the occurrences of 0's in this sequence is 1 - Product_{k>=1} (1 - 1/prime(k)^(A000005(k)+1)) = 0.28458561874647424574... . - 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(A000005(k), e). - Andrew Howroyd, Nov 10 2025

Example

The prime indices of 90 are {1,2,2,3}, with choices:
  ((1),(2),(1,1),(3))
  ((1),(1,1),(2),(3))
  ((1),(2),(1,1),(1,1,1))
  ((1),(1,1),(2),(1,1,1))
so a(90) = 4.

Mathematica

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

Prog

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

Crossrefs

For multiset systems see A355529, set systems A367901.

For not necessarily different choices we have A355731, see A355740.

For divisors instead of constant partitions we have A355739 (also the disjoint case).

For prime factors instead of constant partitions we have A387136.

For all instead of just constant partitions we have A387110, disjoint case A383706.

For initial intervals instead of partitions we have A387111.

For strict instead of constant partitions we have A387115.

Twice partitions of this type are counted by A387179, constant-block case of A296122.

Positions of zero are A387180 (non-choosable), complement A387181 (choosable).

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. A000005, A063834, A261049, A299200, A299201, A335433, A335448, A355741, A357977, A387135.

Sequence in context: A339434 A348325 A103863 * A166395 A378990 A061199

Adjacent sequences: A387117 A387118 A387119 * A387121 A387122 A387123

Keyword

nonn,mult

Author

Gus Wiseman, Aug 26 2025

Status

approved