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Number of strict chains of divisors of n.
13

%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