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

1,6

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.

Links

Table of n, a(n) for n=1..82.

Example

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).

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}]

Crossrefs

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

Sequence in context: A296082 A294786 A294787 * A254578 A348610 A296120

Adjacent sequences: A294785 A294786 A294787 * A294789 A294790 A294791

Keyword

nonn

Author

Gus Wiseman, Nov 08 2017

Status

approved