A357873 Numbers whose multiset of prime factors has all non-isomorphic multiset partitions.

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, 27, 28, 29, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71, 72, 73

Offset: 1

Author

Gus Wiseman, Oct 18 2022

Keywords

• nonn

Comments

These are the positions where A317791 matches A001055.

Examples

			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.
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.
		

Crossrefs

The complement is A357874.

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

A001222 counts prime factors, distinct A001221.

A056239 adds up prime indices, row sums of A112798.

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

Programs

Mathematica

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]}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],UnsameQ@@brute/@mps[primeMS[#]]&]