A381872 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 blocks having a common sum.

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

Offset: 1

Author

Gus Wiseman, Mar 14 2025

Keywords

• nonn

Comments

First differs from A321455 at a(144) = 4, A321455(144) = 3.

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 prime indices of 144 are {1,1,1,1,2,2}, with the following 4 multiset partitions having common block sum:
  {{1,1,1,1,2,2}}
  {{2,2},{1,1,1,1}}
  {{1,1,2},{1,1,2}}
  {{2},{2},{1,1},{1,1}}
with sums: 8, 4, 4, 2, of which 3 are distinct, so a(144) = 3.
The prime indices of 1296 are {1,1,1,1,2,2,2,2}, with the following 7 multiset partitions having common block sum:
  {{1,1,1,1,2,2,2,2}}
  {{2,2,2},{1,1,1,1,2}}
  {{1,1,2,2},{1,1,2,2}}
  {{2,2},{2,2},{1,1,1,1}}
  {{2,2},{1,1,2},{1,1,2}}
  {{1,2},{1,2},{1,2},{1,2}}
  {{2},{2},{2},{2},{1,1},{1,1}}
with sums: 12, 6, 6, 4, 4, 3, 2, of which 5 are distinct, so a(1296) = 5.
		

Crossrefs

With equal blocks instead of sums we have A089723.

Without equal sums we have A317141, before sums A001055, lower A300383.

Positions of terms > 1 are A321454.

Before taking sums we had A321455.

With distinct instead of equal sums we have A381637, before sums A321469.

A000041 counts integer partitions, strict A000009, constant A000005.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A265947 counts refinement-ordered pairs of integer partitions.

Other multiset partitions of prime indices:

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

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

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

Cf. A000720, A000961, A001222, A279787, A279789, A305551, A306017, A321451, A321452, A321453.

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]],SameQ@@Total/@#&]]],{n,100}]