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

1,1

Comments

These are the positions where A317791 differs from A001055.

Links

Table of n, a(n) for n=1..54.

Example

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.

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[#]]&]

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.

Sequence in context: A284787 A214408 A350353 * A325264 A345382 A227680

Adjacent sequences: A357871 A357872 A357873 * A357875 A357876 A357877

Keyword

nonn

Author

Gus Wiseman, Oct 18 2022

Status

approved