A382304 MM-numbers of multiset partitions into sets with a common sum.

1, 2, 3, 4, 5, 8, 9, 11, 13, 16, 17, 25, 27, 29, 31, 32, 41, 43, 47, 59, 64, 67, 73, 79, 81, 83, 101, 109, 113, 121, 125, 127, 128, 137, 139, 143, 149, 157, 163, 167, 169, 179, 181, 191, 199, 211, 233, 241, 243, 256, 257, 269, 271, 277, 283, 289, 293, 313, 317

Offset: 1

Author

Gus Wiseman, Apr 01 2025

Keywords

• nonn

Comments

Also products of prime numbers of squarefree index with a common sum 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}}.

Examples

			The terms together with their prime indices of prime indices begin:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   5: {{2}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  16: {{},{},{},{}}
  17: {{4}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}
		

Links

• Gus Wiseman, Sequences counting and ranking multiset partitions whose part lengths, sums, or averages are constant or strict.

Crossrefs

Set partitions of this type are counted by A035470.

Twice-partitions of this type are counted by A279788.

For just strict blocks we have A302478.

For just a common sum we have A326534, distinct sums A326535.

Factorizations of this type are counted by A382080.

For distinct instead of equal sums we have A382201.

For constant instead of strict blocks we have A382215.

Normal multiset partitions of this type are counted by A382429.

A055396 gives least prime index, greatest A061395.

A056239 adds up prime indices, row sums of A112798.

A058891 counts set-systems, covering A003465, connected A323818.

Cf. A000720, A005117, A050320, A050326, A293511, A302242, A302494, A381635, A381636, A381995.

Programs

Mathematica

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

Formula

Equals A302478 /\ A326534.