%I #7 Aug 24 2020 01:03:21
%S 2,4,4,8,4,12,4,16,8,12,4,32,4,12,12,32,4,32,4,32,12,12,4,80,8,12,16,
%T 32,4,52,4,64,12,12,12,104,4,12,12,80,4,52,4,32,32,12,4,192,8,32,12,
%U 32,4,80,12,80,12,12,4,176,4,12,32,128,12,52,4,32,12,52
%N Number of strict chains of divisors of n.
%F a(n) = A253249(n) + 1.
%e The a(n) chains for n = 1, 2, 4, 6, 8 (empty chains shown as 0):
%e 0 0 0 0 0
%e 1 1 1 1 1
%e 2 2 2 2
%e 2/1 4 3 4
%e 2/1 6 8
%e 4/1 2/1 2/1
%e 4/2 3/1 4/1
%e 4/2/1 6/1 4/2
%e 6/2 8/1
%e 6/3 8/2
%e 6/2/1 8/4
%e 6/3/1 4/2/1
%e 8/2/1
%e 8/4/1
%e 8/4/2
%e 8/4/2/1
%t stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
%t Table[Length[stableSets[Divisors[n],!(Divisible[#1,#2]||Divisible[#2,#1])&]],{n,10}]
%Y A067824 is the case of chains starting with n (or ending with 1).
%Y A074206 is the case of chains from n to 1.
%Y A253249 is the nonempty case.
%Y A000005 counts divisors.
%Y A001055 counts factorizations.
%Y A001222 counts prime factors with multiplicity.
%Y A074206 counts chains of divisors from n to 1.
%Y A122651 counts chains of divisors summing to n.
%Y A167865 counts chains of divisors > 1 summing to n.
%Y A334996 appears to count chains of divisors from n to 1 by length.
%Y A337070 counts chains of divisors starting with A006939(n).
%Y A337071 counts chains of divisors starting with n!.
%Y A337255 counts chains of divisors starting with n by length.
%Y Cf. A001221, A002033, A008480, A124010, A251683, A337105, A337107.
%K nonn
%O 1,1
%A _Gus Wiseman_, Aug 23 2020