A326533 MM-numbers of multiset partitions where each part has a different length.

1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 17, 19, 21, 22, 23, 26, 29, 31, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 53, 57, 58, 59, 61, 62, 65, 67, 69, 70, 71, 73, 74, 77, 78, 79, 82, 83, 86, 87, 89, 94, 95, 97, 101, 103, 106, 107, 109, 111, 113, 114, 115, 118, 119, 122

Offset: 1

Author

Gus Wiseman, Jul 12 2019

Keywords

• nonn

Comments

These are numbers where each prime index has a different Omega (number of prime factors counted with multiplicity). 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. The multiset multisystem with MM-number n is obtained by taking the multiset of prime indices of each prime index of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

Examples

			The sequence of multiset partitions where each part has a different average preceded by their MM-numbers begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  13: {{1,2}}
  14: {{},{1,1}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  26: {{},{1,2}}
  29: {{1,3}}
  31: {{5}}
  34: {{},{4}}
  35: {{2},{1,1}}
		

Links

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

Crossrefs

A subsequence of A005117.

Cf. A007837, A038041, A112798, A302242, A320324, A326026, A326514, A326517, A326534, A326535, A326536, A326537.

Programs

Mathematica

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