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Number of decreasing chains of distinct superior divisors starting with n.
12

%I #13 Jun 24 2021 20:13:17

%S 1,1,1,2,1,2,1,3,2,2,1,5,1,2,2,6,1,5,1,4,2,2,1,11,2,2,3,4,1,7,1,10,2,

%T 2,2,15,1,2,2,10,1,6,1,4,5,2,1,26,2,5,2,4,1,11,2,10,2,2,1,21,1,2,5,20,

%U 2,6,1,4,2,7,1,39,1,2,5,4,2,6,1,23,6,2,1

%N Number of decreasing chains of distinct superior divisors starting with n.

%C We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.

%C These chains have first-quotients (in analogy with first-differences) that are term-wise less than or equal to their decapitation (maximum element removed). Equivalently, x <= y^2 for all adjacent x, y. For example, the divisor chain q = 24/8/4/2 has first-quotients (3,2,2), which are less than or equal to (8,4,2), so q is counted under a(24).

%C Also the number of ordered factorizations of n where each factor is less than or equal to the product of all previous factors.

%H Alois P. Heinz, <a href="/A342085/b342085.txt">Table of n, a(n) for n = 1..65536</a>

%F a(2^n) = A045690(n).

%e The a(n) chains for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:

%e 2 4 8 12 16 20 24 30 32

%e 4/2 8/4 12/4 16/4 20/5 24/6 30/6 32/8

%e 8/4/2 12/6 16/8 20/10 24/8 30/10 32/16

%e 12/4/2 16/4/2 20/10/5 24/12 30/15 32/8/4

%e 12/6/3 16/8/4 24/6/3 30/6/3 32/16/4

%e 16/8/4/2 24/8/4 30/10/5 32/16/8

%e 24/12/4 30/15/5 32/8/4/2

%e 24/12/6 32/16/4/2

%e 24/8/4/2 32/16/8/4

%e 24/12/4/2 32/16/8/4/2

%e 24/12/6/3

%e The a(n) ordered factorizations for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:

%e 2 4 8 12 16 20 24 30 32

%e 2*2 4*2 4*3 4*4 5*4 6*4 6*5 8*4

%e 2*2*2 6*2 8*2 10*2 8*3 10*3 16*2

%e 2*2*3 2*2*4 5*2*2 12*2 15*2 4*2*4

%e 3*2*2 4*2*2 3*2*4 3*2*5 4*4*2

%e 2*2*2*2 4*2*3 5*2*3 8*2*2

%e 4*3*2 5*3*2 2*2*2*4

%e 6*2*2 2*2*4*2

%e 2*2*2*3 4*2*2*2

%e 2*2*3*2 2*2*2*2*2

%e 3*2*2*2

%p a:= proc(n) option remember; 1+add(`if`(d>=n/d,

%p a(d), 0), d=numtheory[divisors](n) minus {n})

%p end:

%p seq(a(n), n=1..128); # _Alois P. Heinz_, Jun 24 2021

%t cmo[n_]:=Prepend[Prepend[#,n]&/@Join@@cmo/@Select[Most[Divisors[n]],#>=n/#&],{n}];

%t Table[Length[cmo[n]],{n,100}]

%Y The restriction to powers of 2 is A045690.

%Y The inferior version is A337135.

%Y The strictly inferior version is A342083.

%Y The strictly superior version is A342084.

%Y The additive version is A342094, with strict case A342095.

%Y The additive version not allowing equality is A342098.

%Y A001055 counts factorizations.

%Y A003238 counts divisibility chains summing to n-1, with strict case A122651.

%Y A038548 counts inferior (or superior) divisors.

%Y A056924 counts strictly inferior (or strictly superior) divisors.

%Y A067824 counts strict chains of divisors starting with n.

%Y A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).

%Y A167865 counts strict chains of divisors > 1 summing to n.

%Y A207375 lists central divisors.

%Y A253249 counts strict chains of divisors.

%Y A334996 counts ordered factorizations by product and length.

%Y A334997 counts chains of divisors of n by length.

%Y - Inferior: A033676, A066839, A072499, A161906.

%Y - Superior: A033677, A070038, A161908, A341676.

%Y - Strictly Inferior: A060775, A070039, A333806, A341674.

%Y - Strictly Superior: A064052/A048098, A140271, A238535, A341673.

%Y Cf. A000203, A001248, A005117, A006530, A020639, A057567, A057568, A112798, A169594, A337105, A342096, A342097.

%K nonn

%O 1,4

%A _Gus Wiseman_, Feb 28 2021