A321469 -id:A321469 - cp's OEIS frontend

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

Gus Wiseman, Mar 14 2025

nonn

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.

			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.

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.

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}]

A382426 MM-numbers of sets of constant multisets with distinct sums.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 38, 41, 42, 46, 51, 53, 55, 57, 59, 62, 66, 67, 69, 77, 82, 83, 85, 93, 95, 97, 102, 103, 106, 109, 110, 114, 115, 118, 119, 123, 127, 131, 133, 134, 138, 154, 155, 157, 159, 161, 165, 166

Offset: 1

Gus Wiseman, Apr 01 2025

nonn

Also products of prime numbers of prime power index 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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  30: {{},{1},{2}}

Twice-partitions of this type are counted by A279786.
For just constant blocks we have A302492.
For just distinct sums we have A326535.
Factorizations of this type are counted by A381635.
For strict instead of constant blocks we have A382201.
Normal multiset partitions of this type are counted by A382203.
For equal instead of distinct sums we have A382215.
An opposite version is A382304.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798.
Cf. A000688, A000720, A000961, A302242, A302494, A321469, A326534, A355743, A356065, A381636, A381716.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@Total/@prix/@prix[#]&&And@@SameQ@@@prix/@prix[#]&]

Equals A302492 /\ A326535.

cp's OEIS Frontend

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.

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