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A357874
Numbers whose multiset of prime factors has at least two multiset partitions that are isomorphic.
1
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.
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.
Sequence in context: A284787 A214408 A350353 * A325264 A345382 A227680
KEYWORD
nonn
AUTHOR
Gus Wiseman, Oct 18 2022
STATUS
approved