%I #10 Mar 26 2020 20:41:59
%S 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 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 2,2,1,0,1,2,0,0,2,2,1,0,5,2,1,0,2,2,2
%N Number of achiral factorizations of n into factors > 1.
%C 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 The a(n) factorizations for n = 2, 6, 27, 36, 243, 216:
%e (2) (6) (27) (36) (243) (216)
%e (2*3) (3*9) (4*9) (3*81) (6*36)
%e (3*3*3) (6*6) (9*27) (8*27)
%e (2*3*6) (3*9*9) (12*18)
%e (2*2*3*3) (3*3*27) (4*6*9)
%e (3*3*3*9) (6*6*6)
%e (3*3*3*3*3) (2*3*36)
%e (2*3*4*9)
%e (2*3*6*6)
%e (2*2*3*3*6)
%e (2*2*2*3*3*3)
%t primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
%t Table[Length[Select[facs[n],Length[graprms[primeMS/@#]]==1&]],{n,100}]
%Y The fully chiral version is A330235.
%Y Planted achiral trees are A003238.
%Y Achiral set-systems are counted by A083323.
%Y BII-numbers of achiral set-systems are A330217.
%Y Non-isomorphic achiral multiset partitions are A330223.
%Y Achiral integer partitions are counted by A330224.
%Y MM-numbers of achiral multisets of multisets are A330232.
%Y Cf. A001055, A007716, A112798, A317533, A330098, A330227, A330228, A330236.
%K nonn
%O 1,4
%A _Gus Wiseman_, Dec 08 2019