A356930 Numbers whose prime indices have all odd prime indices. MM-numbers of finite multisets of finite multisets of odd numbers.

1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 14, 16, 18, 19, 21, 22, 24, 27, 28, 29, 31, 32, 33, 36, 38, 42, 44, 48, 49, 53, 54, 56, 57, 58, 59, 62, 63, 64, 66, 71, 72, 76, 77, 79, 81, 83, 84, 87, 88, 93, 96, 97, 98, 99, 106, 108, 112, 114, 116, 118, 121, 124, 126, 127

Offset: 1

Author

Gus Wiseman, Sep 11 2022

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. We define the multiset of multisets with MM-number n to be formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. The combined size of this multiset of multisets is A302242(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 initial terms and corresponding multisets of multisets:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  11: {{3}}
  12: {{},{},{1}}
  14: {{},{1,1}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  24: {{},{},{},{1}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  29: {{1,3}}
  31: {{5}}
  32: {{},{},{},{},{}}
		

Links

• Gus Wiseman, Counting and ranking classes of multiset partitions related to gapless multisets

Crossrefs

Multisets of odd numbers are counted by A000009, ranked by A066208.

Factorizations of this type are counted by A356931.

The version for odd lengths instead of parts is A356935, ranked by A089259.

Other conditions: A302478, A302492, A356939, A356940, A356944, A356955.

A000041 counts integer partitions, strict A000009.

A000688 counts factorizations into prime powers.

A001055 counts factorizations.

A001221 counts prime divisors, sum A001414.

A001222 counts prime factors with multiplicity.

A056239 adds up prime indices, row sums of A112798.

Cf. A000720, A003963, A011782, A055887, A076610, A302242, A304969.

Programs

Mathematica

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