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%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