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

1,2

Comments

These are the positions where A317791 matches A001055.

Links

Table of n, a(n) for n=1..67.

Example

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.

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 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.

Sequence in context: A004441 A004438 A109425 * A226537 A349294 A153679

Adjacent sequences: A357870 A357871 A357872 * A357874 A357875 A357876

Keyword

nonn

Author

Gus Wiseman, Oct 18 2022

Status

approved