A381454 Number of multisets that can be obtained by choosing a strict integer partition of each prime index of n and taking the multiset union.

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

Offset: 1

Author

Gus Wiseman, Mar 08 2025

Keywords

• nonn

Comments

First differs from A357982 at a(25) = 3, A357982(25) = 4.

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.

A multiset partition can be regarded as an arrow in the ranked poset of integer partitions. For example, we have {{1},{1,2},{1,3},{1,2,3}}: {1,1,1,1,2,2,3,3} -> {1,3,4,6}, or (33221111) -> (6431) (depending on notation).

Set multipartitions are generally not transitive. For example, we have arrows: {{1},{1,2}}: {1,1,2} -> {1,3} and {{1,3}}: {1,3} -> {4}, but there is no set multipartition {1,1,2} -> {4}.

Examples

			The a(25) = 3 multisets are: {3,3}, {1,2,3}, {1,1,2,2}.
		

Crossrefs

For constant instead of strict partitions see A381453, A355733, A381455, A000688.

Positions of 1 are A003586.

The upper version is A381078, before sums A050320.

For distinct block-sums see A381634, A381633, A381806.

Multiset partitions of prime indices:

- For multiset partitions (A001055) see A317141 (upper), A300383 (lower).

- For strict multiset partitions (A045778) see A381452.

- For set systems (A050326, zeros A293243) see A381441 (upper).

- For sets of constant multisets (A050361) see A381715.

- For strict multiset partitions with distinct sums (A321469) see A381637.

- For sets of constant multisets with distinct sums (A381635, zeros A381636) see A381716.

More on set systems: A050342, A116539, A296120, A318361.

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.

More on set multipartitions with distinct sums: A279785, A381717, A381718.

A000041 counts integer partitions, strict A000009.

A000040 lists the primes.

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.

A265947 counts refinement-ordered pairs of integer partitions.

A358914 counts twice-partitions into distinct strict partitions.

Cf. A000720, A001222, A002846, A005117, A066328, A213242, A213385, A213427, A293511, A299200, A299201, A299202, A300385, A317142.

Programs

Mathematica

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

Formula

a(A002110(n)) = A381808(n).