A357874 Numbers whose multiset of prime factors has at least two multiset partitions that are isomorphic.

30, 36, 42, 60, 66, 70, 78, 84, 90, 100, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 196, 198, 204, 210, 216, 220, 222, 225, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270

Offset: 1

Author

Gus Wiseman, Oct 18 2022

Keywords

• nonn

Comments

These are the positions where A317791 differs from A001055.

Examples

			The terms together with their prime indices begin:
   30: {1,2,3}
   36: {1,1,2,2}
   42: {1,2,4}
   60: {1,1,2,3}
   66: {1,2,5}
   70: {1,3,4}
   78: {1,2,6}
   84: {1,1,2,4}
   90: {1,2,2,3}
  100: {1,1,3,3}
For example, the multiset partitions of the prime indices of 36 include {{1},{1,2,2}} and {{2},{1,1,2}}, which are isomorphic, so 36 is in the sequence.
		

Crossrefs

The complement is A357873.

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, A300913, 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[#]]&]