A294788 Number of twice-factorizations of type (Q,P,Q) and product n.

1, 1, 1, 1, 1, 3, 1, 3, 1, 3, 1, 5, 1, 3, 3, 3, 1, 5, 1, 5, 3, 3, 1, 12, 1, 3, 3, 5, 1, 12, 1, 5, 3, 3, 3, 13, 1, 3, 3, 12, 1, 12, 1, 5, 5, 3, 1, 19, 1, 5, 3, 5, 1, 12, 3, 12, 3, 3, 1, 26, 1, 3, 5, 11, 3, 12, 1, 5, 3, 12, 1, 26, 1, 3, 5, 5, 3, 12, 1, 19, 3, 3

Offset: 1

Author

Gus Wiseman, Nov 08 2017

Keywords

• nonn

Comments

a(n) is the number of ways to choose a product-preserving permutation of a set partition of a factorization of n into distinct factors greater than one.

Examples

			The a(36) = 13 twice-factorizations are: (2)*(3)*(6), (2)*(3*6), (6)*(2*3), (2*3)*(6), (2*6)*(3), (2*3*6), (2)*(18), (2*18), (3)*(12), (3*12), (4)*(9), (4*9), (36).
		

Crossrefs

Cf. A001055, A063834, A279790, A281113, A294786, A294787.

Programs

Mathematica

nn=100;
sfs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sfs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Total[Sum[Times@@Factorial/@Length/@Split[Sort[Times@@@f]],{f,sps[Sort[#]]}]&/@sfs[n]],{n,nn}]