This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A321468 #8 Nov 13 2018 12:55:05
%S A321468 1,1,1,1,2,2,4,4,10,20,40,40,116,116,232,464,1440,1440,4192,4192,
%T A321468 11640,23280,46560,46560,157376
%N 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.
%C A321468 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 A321468 The a(2) = 1 through a(8) = 10 factorizations:
%e A321468 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 A321468 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 A321468 2*2*3*3*4*5 2*2*3*3*4*5*7 2*2*3*3*4*5*7*8
%e A321468 2*2*2*2*3*3*5 2*2*2*2*3*3*5*7 2*2*3*4*4*5*6*7
%e A321468 2*2*2*2*3*3*5*7*8
%e A321468 2*2*2*2*3*4*5*6*7
%e A321468 2*2*2*3*3*4*4*5*7
%e A321468 2*2*2*2*2*2*3*5*6*7
%e A321468 2*2*2*2*2*3*3*4*5*7
%e A321468 2*2*2*2*2*2*2*3*3*5*7
%e A321468 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 A321468 facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
%t A321468 Table[Length[Union[Sort/@Join@@@Tuples[facs/@Range[2,n]]]],{n,10}]
%Y A321468 Dominated by A321514.
%Y A321468 Cf. A001055, A066723, A076716, A157612, A242422, A265947, A300383, A317144, A317145, A317534, A321467, A321470, A321471, A321472.
%K A321468 nonn,more
%O A321468 0,5
%A A321468 _Gus Wiseman_, Nov 11 2018