login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Squarefree numbers whose prime indices have no equivalent primes.
1

%I #5 Dec 31 2018 13:17:53

%S 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,

%T 46,51,53,55,57,59,61,62,65,66,67,69,70,71,74,77,78,82,83,85,87,89,91,

%U 93,95,97,102,103,105,106,107,109,110,111,114,115,118,119

%N Squarefree numbers whose prime indices have no equivalent primes.

%C 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.

%C 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.

%C 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).

%e The sequence of all strict T_0 multiset multisystems together with their MM-numbers begins:

%e 1: {}

%e 2: {{}}

%e 3: {{1}}

%e 5: {{2}}

%e 6: {{},{1}}

%e 7: {{1,1}}

%e 10: {{},{2}}

%e 11: {{3}}

%e 14: {{},{1,1}}

%e 15: {{1},{2}}

%e 17: {{4}}

%e 19: {{1,1,1}}

%e 21: {{1},{1,1}}

%e 22: {{},{3}}

%e 23: {{2,2}}

%e 30: {{},{1},{2}}

%e 31: {{5}}

%e 33: {{1},{3}}

%e 34: {{},{4}}

%e 35: {{2},{1,1}}

%e 37: {{1,1,2}}

%e 38: {{},{1,1,1}}

%e 39: {{1},{1,2}}

%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];

%t dual[eds_]:=Table[First/@Position[eds,x],{x,Union@@eds}];

%t Select[Range[100],And[SquareFreeQ[#],UnsameQ@@dual[primeMS/@primeMS[#]]]&]

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 28 2018