A381635 Number of ways to partition the prime indices of n into constant blocks with distinct sums.

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

Offset: 1

Author

Gus Wiseman, Mar 09 2025

Keywords

• nonn

Comments

First differs from A381716 at a(1728) = 5, A381716(1728) = 4.

Also the number of factorizations on n into prime powers > 1 with distinct sums of prime indices (A056239).

Examples

			The a(432) = 3 multiset partitions:
  {{2,2,2},{1,1,1,1}}
  {{1},{1,1,1},{2,2,2}}
  {{1},{2},{2,2},{1,1,1}}
Note {{2},{2,2},{1,1,1,1}} is not included, as it does not have distinct block-sums.
		

Crossrefs

Without distinct block-sums we have A000688, after sums A381455 (upper), A381453 (lower).

For distinct blocks instead of sums we have A050361, after sums A381715.

For strict instead of constant we have A381633 (zeros A381806), after sums A381634.

Positions of 0 are A381636.

Taking block-sums (and sorting) gives A381716.

Other multiset partitions of prime indices:

More on multiset partitions into constant blocks: A006171, A279784, A295935.

A001055 counts multiset partitions, 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, A213385, A293511, A299202, A300385, A317142, A381870.

Programs

Mathematica

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