A381715 Number of multisets that can be obtained by taking the sum of each block of a multiset partition of the prime indices of n into distinct constant blocks.

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

Offset: 1

Author

Gus Wiseman, Mar 10 2025

Keywords

• nonn

Comments

First differs from A050361 at a(1728) = 7, A050361(1728) = 8.

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.

Examples

			The prime indices of 1728 are {1,1,1,1,1,1,2,2,2}, with multiset partitions into distinct constant blocks:
  {{2,2,2},{1,1,1,1,1,1}}
  {{1},{2,2,2},{1,1,1,1,1}}
  {{2},{2,2},{1,1,1,1,1,1}}
  {{1,1},{2,2,2},{1,1,1,1}}
  {{1},{2},{2,2},{1,1,1,1,1}}
  {{1},{1,1},{1,1,1},{2,2,2}}
  {{2},{1,1},{2,2},{1,1,1,1}}
  {{1},{2},{1,1},{2,2},{1,1,1}}
with sums:
  {6,6}
  {1,5,6}
  {2,4,6}
  {2,4,6}
  {1,2,4,5}
  {1,2,3,6}
  {2,2,4,4}
  {1,2,2,3,4}
of which 7 are distinct, so a(1728) = 7.
		

Crossrefs

Without distinct blocks (A000688) we have A381455, lower (A355731) A381453.

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

Positions of terms > 1 are A046099.

Before taking sums we had A050361.

For equal instead of distinct blocks we have A362421.

For strict instead of constant blocks we have A381441, before sums A050326.

For just distinct blocks we have A381452, before sums A045778.

For distinct sums we have A381716, before sums A381635, zeros A381636.

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.

Cf. A000720, A001222, A002846, A005117, A050342, A213242, A213385, A293511, A299202, A300385, A317142, A381870.

Programs

Mathematica

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Union[Sort[Total/@#]&/@Select[mps[prix[n]],UnsameQ@@#&&And@@SameQ@@@#&]]],{n,100}]