cp's OEIS Frontend

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

%I A357874 #6 Oct 18 2022 13:32:06
%S A357874 30,36,42,60,66,70,78,84,90,100,102,105,110,114,120,126,130,132,138,
%T A357874 140,150,154,156,165,168,170,174,180,182,186,190,195,196,198,204,210,
%U A357874 216,220,222,225,228,230,231,234,238,240,246,252,255,258,260,264,266,270
%N A357874 Numbers whose multiset of prime factors has at least two multiset partitions that are isomorphic.
%C A357874 These are the positions where A317791 differs from A001055.
%e A357874 The terms together with their prime indices begin:
%e A357874    30: {1,2,3}
%e A357874    36: {1,1,2,2}
%e A357874    42: {1,2,4}
%e A357874    60: {1,1,2,3}
%e A357874    66: {1,2,5}
%e A357874    70: {1,3,4}
%e A357874    78: {1,2,6}
%e A357874    84: {1,1,2,4}
%e A357874    90: {1,2,2,3}
%e A357874   100: {1,1,3,3}
%e A357874 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.
%t A357874 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]}];
%t A357874 mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
%t A357874 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A357874 Select[Range[100],!UnsameQ@@brute/@mps[primeMS[#]]&]
%Y A357874 The complement is A357873.
%Y A357874 A001055 counts multiset partitions of prime indices, non-isomorphic A317791.
%Y A357874 A001222 counts prime factors, distinct A001221.
%Y A357874 A056239 adds up prime indices, row sums of A112798.
%Y A357874 Cf. A000612, A007716, A055621, A283877, A300913, A302545, A317533, A321194.
%K A357874 nonn
%O A357874 1,1
%A A357874 _Gus Wiseman_, Oct 18 2022