A322846 Squarefree numbers whose prime indices have no equivalent primes.

1, 2, 3, 5, 6, 7, 10, 11, 14, 15, 17, 19, 21, 22, 23, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 46, 51, 53, 55, 57, 59, 61, 62, 65, 66, 67, 69, 70, 71, 74, 77, 78, 82, 83, 85, 87, 89, 91, 93, 95, 97, 102, 103, 105, 106, 107, 109, 110, 111, 114, 115, 118, 119

Offset: 1

Author

Gus Wiseman, Dec 28 2018

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.

In an integer partition, two primes are equivalent if each part has in its prime factorization the same multiplicity of both primes. For example, in (6,5) the primes {2,3} are equivalent while {2,5} and {3,5} are not. In (30,6) also, the primes {2,3} are equivalent, while {2,5} and {3,5} are not.

Also MM-numbers of strict T_0 multiset multisystems. A multiset multisystem is a finite multiset of finite multisets. The multiset multisystem 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 multisystem with MM-number 78 is {{},{1},{1,2}}. The dual of a multiset multisystem has, for each vertex, one block consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}. The T_0 condition means the dual is strict (no repeated parts).

Examples

			The sequence of all strict T_0 multiset multisystems together with their MM-numbers begins:
   1: {}
   2: {{}}
   3: {{1}}
   5: {{2}}
   6: {{},{1}}
   7: {{1,1}}
  10: {{},{2}}
  11: {{3}}
  14: {{},{1,1}}
  15: {{1},{2}}
  17: {{4}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  22: {{},{3}}
  23: {{2,2}}
  30: {{},{1},{2}}
  31: {{5}}
  33: {{1},{3}}
  34: {{},{4}}
  35: {{2},{1,1}}
  37: {{1,1,2}}
  38: {{},{1,1,1}}
  39: {{1},{1,2}}
		

Crossrefs

Cf. A000009, A005117, A056239, A059201, A112798, A302242, A302505, A316978, A316979, A316983, A319558, A319564, A319728, A322847.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];
Select[Range[100],And[SquareFreeQ[#],UnsameQ@@dual[primeMS/@primeMS[#]]]&]