A321468 - OEIS

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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

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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