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

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, 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, 2, 2, 1, 0, 1, 2, 0, 0, 2, 2, 1, 0, 5, 2, 1, 0, 2, 2, 2

Offset: 1

Author

Gus Wiseman, Dec 08 2019

Keywords

• nonn

Comments

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.

Examples

			The a(n) factorizations for n = 2, 6, 27, 36, 243, 216:
  (2)  (6)    (27)     (36)       (243)        (216)
       (2*3)  (3*9)    (4*9)      (3*81)       (6*36)
              (3*3*3)  (6*6)      (9*27)       (8*27)
                       (2*3*6)    (3*9*9)      (12*18)
                       (2*2*3*3)  (3*3*27)     (4*6*9)
                                  (3*3*3*9)    (6*6*6)
                                  (3*3*3*3*3)  (2*3*36)
                                               (2*3*4*9)
                                               (2*3*6*6)
                                               (2*2*3*3*6)
                                               (2*2*2*3*3*3)
		

Crossrefs

The fully chiral version is A330235.

Planted achiral trees are A003238.

Achiral set-systems are counted by A083323.

BII-numbers of achiral set-systems are A330217.

Non-isomorphic achiral multiset partitions are A330223.

Achiral integer partitions are counted by A330224.

MM-numbers of achiral multisets of multisets are A330232.

Cf. A001055, A007716, A112798, A317533, A330098, A330227, A330228, A330236.

Programs

Mathematica

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]],i},{i,Length[p]}])],{p,Permutations[Union@@m]}]];
Table[Length[Select[facs[n],Length[graprms[primeMS/@#]]==1&]],{n,100}]