A381078 Number of multisets that can be obtained by partitioning the prime indices of n into a multiset of sets (set multipartition) and taking their sums.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 5, 1, 1, 2, 2, 2, 3, 1, 2, 2, 2, 1, 5, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 1, 6, 1, 2, 2, 1, 2, 5, 1, 2, 2, 5, 1, 3, 1, 2, 2, 2, 2, 5, 1, 2, 1, 2, 1, 6, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Mar 05 2025

Keywords

• nonn

Comments

First differs from A050320 at a(210) = 13, A050320(210) = 15. This comes from the set multipartitions {{3},{1,2,4}} and {{1,2},{3,4}}, and from {{4},{1,2,3}} and {{1,3},{2,4}}.

Also the number of multisets that can be obtained by taking the sums of prime indices of each factor in a factorization of n into squarefree numbers > 1.

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 prime indices of 60 are {1,1,2,3}, with set multipartitions:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}
with block-sums: {1,6}, {3,4}, {1,1,5}, {1,2,4}, {1,3,3}, {1,1,2,3}, which are all different multisets, so a(60) = 6.
		

Links

• Robert Price, Table of n, a(n) for n = 1..1000

Crossrefs

Before taking sums we had A050320, strict A050326 (zeros A293243), distinct sums A381633.

For distinct blocks we have A381441.

The lower version is A381454.

For distinct block-sums we have A381634.

Other multiset partitions of prime indices:

- For multisets of constant multisets (A000688) see A381455 (upper), A381453 (lower).

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

- For strict multiset partitions (A045778) see A381452.

- For sets of constant multisets (A050361) see A381717.

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

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

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

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.

Cf. A000720, A001222, A002846, A005117, A025487, A066328, A213242, A213385, A213427, A299201, A299202, A300385.

Programs

Mathematica

hwt[n_]:=Total[Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]*k]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[(Prepend[#,d]&)/@Select[sqfacs[n/d],Min@@#>=d&],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[Union[Sort[hwt/@#]&/@sqfacs[n]]],{n,100}]

Formula

a(A002110(n)) = A066723(n).