login
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.
6
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.
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}]
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 11 2018
STATUS
approved