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

1,2

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}}.

Links

Table of n, a(n) for n=1..64.

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

Example

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}}

Mathematica

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

Crossrefs

A subsequence of A005117.

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

Sequence in context: A325362 A339563 A325396 * A144147 A068422 A085118

Adjacent sequences: A326530 A326531 A326532 * A326534 A326535 A326536

Keyword

nonn

Author

Gus Wiseman, Jul 12 2019

Status

approved