A368603 Products of odd primes of squarefree index. MM-numbers of set multipartitions.

1, 3, 5, 9, 11, 13, 15, 17, 25, 27, 29, 31, 33, 39, 41, 43, 45, 47, 51, 55, 59, 65, 67, 73, 75, 79, 81, 83, 85, 87, 93, 99, 101, 109, 113, 117, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 149, 153, 155, 157, 163, 165, 167, 169, 177, 179, 181, 187

Offset: 1

Author

Gus Wiseman, Jan 08 2024

Keywords

• nonn

Comments

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 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}}.

A set multipartition is a finite multiset of finite nonempty sets.

Examples

			The terms together with the corresponding set multipartitions begin:
   1: {}
   3: {{1}}
   5: {{2}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
		

Crossrefs

Odd case of A302478.

Products of odd terms of A302491.

A049311 counts non-isomorphic set multipartitions, strict A283877.

A050320 counts set multipartitions of prime indices.

A056239 adds up prime indices, row sums of A112798.

A089259 counts set multipartitions of integer partitions.

A116540 counts set multipartitions covering an initial interval by weight.

A368533 lists numbers with squarefree binary indices.

Cf. A000040, A000720, A001222, A005117, A006450, A076610, A270995, A296119, A302242, A302590, A339113.

Programs

Mathematica

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