cp's OEIS Frontend

A330234 Number of achiral factorizations of n into factors > 1.

%I A330234 #10 Mar 26 2020 20:41:59
%S A330234 1,1,1,2,1,2,1,3,2,2,1,0,1,2,2,5,1,0,1,0,2,2,1,0,2,2,3,0,1,2,1,7,2,2,
%T A330234 2,5,1,2,2,0,1,2,1,0,0,2,1,0,2,0,2,0,1,0,2,0,2,2,1,0,1,2,0,11,2,2,1,0,
%U A330234 2,2,1,0,1,2,0,0,2,2,1,0,5,2,1,0,2,2,2
%N A330234 Number of achiral factorizations of n into factors > 1.
%C A330234 A multiset of multisets is achiral if it is not changed by any permutation of the vertices. A factorization is achiral if taking the multiset of prime indices of each factor gives an achiral multiset of multisets.
%e A330234 The a(n) factorizations for n = 2, 6, 27, 36, 243, 216:
%e A330234   (2)  (6)    (27)     (36)       (243)        (216)
%e A330234        (2*3)  (3*9)    (4*9)      (3*81)       (6*36)
%e A330234               (3*3*3)  (6*6)      (9*27)       (8*27)
%e A330234                        (2*3*6)    (3*9*9)      (12*18)
%e A330234                        (2*2*3*3)  (3*3*27)     (4*6*9)
%e A330234                                   (3*3*3*9)    (6*6*6)
%e A330234                                   (3*3*3*3*3)  (2*3*36)
%e A330234                                                (2*3*4*9)
%e A330234                                                (2*3*6*6)
%e A330234                                                (2*2*3*3*6)
%e A330234                                                (2*2*2*3*3*3)
%t A330234 primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A330234 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A330234 graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t A330234 Table[Length[Select[facs[n],Length[graprms[primeMS/@#]]==1&]],{n,100}]
%Y A330234 The fully chiral version is A330235.
%Y A330234 Planted achiral trees are A003238.
%Y A330234 Achiral set-systems are counted by A083323.
%Y A330234 BII-numbers of achiral set-systems are A330217.
%Y A330234 Non-isomorphic achiral multiset partitions are A330223.
%Y A330234 Achiral integer partitions are counted by A330224.
%Y A330234 MM-numbers of achiral multisets of multisets are A330232.
%Y A330234 Cf. A001055, A007716, A112798, A317533, A330098, A330227, A330228, A330236.
%K A330234 nonn
%O A330234 1,4
%A A330234 _Gus Wiseman_, Dec 08 2019