A321468 Number of factorizations of n! into factors > 1 that can be obtained by taking the multiset union of a choice of factorizations of each positive integer from 2 to n into factors > 1.

1, 1, 1, 1, 2, 2, 4, 4, 10, 20, 40, 40, 116, 116, 232, 464, 1440, 1440, 4192, 4192, 11640, 23280, 46560, 46560, 157376

Offset

0,5

Comments

a(n) is the number of factorizations finer than (2*3*...*n) in the poset of factorizations of n! into factors > 1, ordered by refinement.

Links

Table of n, a(n) for n=0..24.

Example

The a(2) = 1 through a(8) = 10 factorizations:
2  2*3  2*3*4    2*3*4*5    2*3*4*5*6      2*3*4*5*6*7      2*3*4*5*6*7*8
        2*2*2*3  2*2*2*3*5  2*2*2*3*5*6    2*2*2*3*5*6*7    2*2*2*3*5*6*7*8
                            2*2*3*3*4*5    2*2*3*3*4*5*7    2*2*3*3*4*5*7*8
                            2*2*2*2*3*3*5  2*2*2*2*3*3*5*7  2*2*3*4*4*5*6*7
                                                            2*2*2*2*3*3*5*7*8
                                                            2*2*2*2*3*4*5*6*7
                                                            2*2*2*3*3*4*4*5*7
                                                            2*2*2*2*2*2*3*5*6*7
                                                            2*2*2*2*2*3*3*4*5*7
                                                            2*2*2*2*2*2*2*3*3*5*7
For example, 2*2*2*2*2*2*3*5*6*7 = (2)*(3)*(2*2)*(5)*(6)*(7)*(2*2*2), so (2*2*2*2*2*2*3*5*6*7) is counted under a(8).

Mathematica

facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
Table[Length[Union[Sort/@Join@@@Tuples[facs/@Range[2, n]]]], {n, 10}]

Crossrefs

Dominated by A321514.

Cf. A001055, A066723, A076716, A157612, A242422, A265947, A300383, A317144, A317145, A317534, A321467, A321470, A321471, A321472.

Sequence in context: A329394 A057784 A209284 * A288044 A081164 A125553

Adjacent sequences: A321465 A321466 A321467 * A321469 A321470 A321471

Keyword

nonn,more

Author

Gus Wiseman, Nov 11 2018

Status

approved