A321468 - OEIS

%I #8 Nov 13 2018 12:55:05

%S 1,1,1,1,2,2,4,4,10,20,40,40,116,116,232,464,1440,1440,4192,4192,

%T 11640,23280,46560,46560,157376

%N 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.

%C 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.

%e The a(2) = 1 through a(8) = 10 factorizations:

%e 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

%e         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

%e                             2*2*3*3*4*5    2*2*3*3*4*5*7    2*2*3*3*4*5*7*8

%e                             2*2*2*2*3*3*5  2*2*2*2*3*3*5*7  2*2*3*4*4*5*6*7

%e                                                             2*2*2*2*3*3*5*7*8

%e                                                             2*2*2*2*3*4*5*6*7

%e                                                             2*2*2*3*3*4*4*5*7

%e                                                             2*2*2*2*2*2*3*5*6*7

%e                                                             2*2*2*2*2*3*3*4*5*7

%e                                                             2*2*2*2*2*2*2*3*3*5*7

%e 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).

%t facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];

%t Table[Length[Union[Sort/@Join@@@Tuples[facs/@Range[2,n]]]],{n,10}]

%Y Dominated by A321514.

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

%K nonn,more

%O 0,5

%A _Gus Wiseman_, Nov 11 2018