A357873 - OEIS

%I #7 Oct 18 2022 13:32:12

%S 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,

%T 27,28,29,31,32,33,34,35,37,38,39,40,41,43,44,45,46,47,48,49,50,51,52,

%U 53,54,55,56,57,58,59,61,62,63,64,65,67,68,69,71,72,73

%N Numbers whose multiset of prime factors has all non-isomorphic multiset partitions.

%C These are the positions where A317791 matches A001055.

%e The multiset partitions of the prime indices of 12 are: {{1,1,2}}, {{1},{1,2}}, {{1,1},{2}}, {{1},{1},{2}}, all of which are non-isomorphic, so 12 is in the sequence.

%e The multiset partitions of the prime indices of 30 are: {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}, of which the middle 3 are isomorphic, so 30 is not in the sequence.

%t brute[m_]:=If[Union@@m!={}&&Union@@m!=Range[Max@@Flatten[m]],brute[m/.Rule@@@Table[{(Union@@m)[[i]],i},{i,Length[Union@@m]}]],First[Sort[brute[m,1]]]];brute[m_,1]:=Table[Sort[Sort/@(m/.Rule@@@Table[{i,p[[i]]},{i,Length[p]}])],{p,Permutations[Union@@m]}];

%t mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];

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

%t Select[Range[100],UnsameQ@@brute/@mps[primeMS[#]]&]

%Y The complement is A357874.

%Y A001055 counts multiset partitions of prime indices, non-isomorphic A317791.

%Y A001222 counts prime factors, distinct A001221.

%Y A056239 adds up prime indices, row sums of A112798.

%Y Cf. A000612, A007716, A055621, A283877, A302545, A317533, A321194.

%K nonn

%O 1,2

%A _Gus Wiseman_, Oct 18 2022