OFFSET
0,4
COMMENTS
a(n) is the number of factorizations coarser than (2*3*...*n) in the poset of factorizations of n! into factors > 1, ordered by refinement.
EXAMPLE
The a(1) = 1 through a(5) = 15 factorizations:
() (2) (6) (24) (120)
(2*3) (3*8) (2*60)
(4*6) (3*40)
(2*12) (4*30)
(2*3*4) (5*24)
(6*20)
(8*15)
(10*12)
(3*5*8)
(4*5*6)
(2*3*20)
(2*4*15)
(2*5*12)
(3*4*10)
(2*3*4*5)
For example, 10*12 = (2*5)*(3*4), so (10*12) is counted under a(5).
MATHEMATICA
sps[{}]:={{}}; sps[set:{i_, ___}]:=Join@@Function[s, Prepend[#, s]&/@sps[Complement[set, s]]]/@Cases[Subsets[set], {i, ___}];
Table[Length[Union[Sort/@Apply[Times, sps[Range[2, n]], {2}]]], {n, 10}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Gus Wiseman, Nov 11 2018
STATUS
approved