A382525 Number of times n appears in A048767 (rank of Look-and-Say partition of prime indices). Number of ordered set partitions whose block-sums are the prime signature of n.

1, 1, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 3, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 4, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 2, 0, 1, 0, 0, 0, 0

Offset: 1

Author

Gus Wiseman, Apr 05 2025

Keywords

• nonn

Comments

The Look-and-Say partition of a multiset or partition y is obtained by interchanging parts with multiplicities. Hence, the multiplicity of k in the Look-and-Say partition of y is the sum of all parts that appear exactly k times. For example, starting with (3,2,2,1,1) we get (2,2,2,1,1,1), the multiset union of ((1,1,1),(2,2),(2)).

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, sum A056239.

Also the number of ways to choose a set of disjoint strict integer partitions, one of each nonzero multiplicity in the prime factorization of n.

Examples

			The a(27) = 2 partitions with Look-and-Say partition (2,2,2) are: (3,3), (2,2,1,1).
The prime indices of 3456 are {1,1,1,1,1,1,1,2,2,2}, and the partitions with Look-and-Say partition (2,2,2,1,1,1,1,1,1,1) are:
  (7,3,3)
  (7,2,2,1,1)
  (6,3,3,1)
  (5,3,3,2)
  (4,3,3,2,1)
  (4,3,2,2,1,1)
so a(3456) = 6.
		

Crossrefs

Positions of positive terms are A351294, conjugate A381432.

Positions of 0 are A351295, conjugate A381433.

Positions of 1 are A381540, conjugate A381434.

Positions of terms > 1 are A381541, conjugate A381435.

Positions of first appearances are A382775.

A000670 counts ordered set partitions.

A003963 gives product of prime indices.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A122111 represents conjugation in terms of Heinz numbers.

A239455 counts Look-and-Say partitions, complement A351293.

A381436 lists the section-sum partition of prime indices, ranks A381431.

A381440 lists the Look-and-Say partition of prime indices, ranks A048767.

Cf. A003557, A047966, A048768, A050361, A051903, A051904, A066328, A071178, A116861, A130091, A217605, A239964.

Programs

Mathematica

stp[y_]:=Select[Tuples[Select[IntegerPartitions[#],UnsameQ@@#&]&/@y],UnsameQ@@Join@@#&];
Table[Length[stp[Last/@FactorInteger[n]]],{n,100}]

Formula

a(2^n) = A000009(n).

a(prime(n)) = 1.