A322847 - OEIS

%I #5 Dec 31 2018 13:18:01

%S 1,2,3,4,5,6,7,8,9,10,11,12,14,15,16,17,18,19,20,21,22,23,24,25,27,28,

%T 30,31,32,33,34,35,36,37,38,39,40,41,42,44,45,46,48,49,50,51,53,54,55,

%U 56,57,59,60,61,62,63,64,65,66,67,68,69,70,71,72,74,75

%N Numbers whose prime indices have no equivalent primes.

%C The complement is {13, 26, 29, 43, 47, 52, 58, 73, 79, 86, 94, ...}.

%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 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 prime indices of 339 are {2, 30}, in which the primes {3,5} are equivalent, so 339 is not in the sequence.

%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],UnsameQ@@dual[primeMS/@primeMS[#]]&]

%Y Cf. A056239, A059201, A112798, A302242, A316978, A316979, A316983, A319558, A319564, A319728, A322846.

%K nonn

%O 1,2

%A _Gus Wiseman_, Dec 28 2018