A381633 Number of ways to partition the prime indices of n into sets with distinct sums.

1, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 0, 0, 2, 0, 1, 1, 4, 1, 0, 2, 2, 2, 1, 1, 2, 2, 0, 1, 5, 1, 1, 1, 2, 1, 0, 0, 1, 2, 1, 1, 0, 2, 0, 2, 2, 1, 3, 1, 2, 1, 0, 2, 5, 1, 1, 2, 4, 1, 0, 1, 2, 1, 1, 2, 5, 1, 0, 0, 2, 1, 4, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Mar 09 2025

Keywords

• nonn

Comments

First differs from A050326 at 30, 60, 70, 90, ...

First differs from A339742 at 42, 66, 78, 84, ...

First differs from A381634 at a(210) = 12, A381634(210) = 10.

Also the number of factorizations on n into squarefree numbers > 1 with distinct sums of prime indices.

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.

Examples

			The A050320(60) = 6 ways to partition {1,1,2,3} into sets are:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{1},{2,3}}
  {{1},{2},{1,3}}
  {{1},{3},{1,2}}
  {{1},{1},{2},{3}}
Of these, only the following have distinct block-sums:
  {{1},{1,2,3}}
  {{1,2},{1,3}}
  {{1},{2},{1,3}}
So a(60) = 3.
		

Crossrefs

Without distinct block-sums we have A050320, after sums A381078 (lower A381454).

For distinct blocks instead of sums we have A050326, after sums A381441, see A358914.

Taking block-sums (and sorting) gives A381634.

For constant instead of strict blocks we have A381635, see A381716, A381636.

Positions of 0 are A381806, superset of A293243.

Positions of 1 are A381870, superset of A293511.

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

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

A000041 counts integer partitions, strict A000009.

A001055 count multiset partitions of prime indices, see A317141 (upper), A300383 (lower).

A003963 gives product of prime indices.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A265947 counts refinement-ordered pairs of integer partitions.

Cf. A000720, A001222, A002846, A005117, A213242, A299202, A300385, A317142, A317143, A321469.

Programs

Mathematica

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