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 A342085 #13 Jun 24 2021 20:13:17
%S A342085 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 A342085 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 A342085 2,6,1,4,2,7,1,39,1,2,5,4,2,6,1,23,6,2,1
%N A342085 Number of decreasing chains of distinct superior divisors starting with n.
%C A342085 We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.
%C A342085 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 A342085 Also the number of ordered factorizations of n where each factor is less than or equal to the product of all previous factors.
%H A342085 Alois P. Heinz, <a href="/A342085/b342085.txt">Table of n, a(n) for n = 1..65536</a>
%F A342085 a(2^n) = A045690(n).
%e A342085 The a(n) chains for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
%e A342085 2 4 8 12 16 20 24 30 32
%e A342085 4/2 8/4 12/4 16/4 20/5 24/6 30/6 32/8
%e A342085 8/4/2 12/6 16/8 20/10 24/8 30/10 32/16
%e A342085 12/4/2 16/4/2 20/10/5 24/12 30/15 32/8/4
%e A342085 12/6/3 16/8/4 24/6/3 30/6/3 32/16/4
%e A342085 16/8/4/2 24/8/4 30/10/5 32/16/8
%e A342085 24/12/4 30/15/5 32/8/4/2
%e A342085 24/12/6 32/16/4/2
%e A342085 24/8/4/2 32/16/8/4
%e A342085 24/12/4/2 32/16/8/4/2
%e A342085 24/12/6/3
%e A342085 The a(n) ordered factorizations for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
%e A342085 2 4 8 12 16 20 24 30 32
%e A342085 2*2 4*2 4*3 4*4 5*4 6*4 6*5 8*4
%e A342085 2*2*2 6*2 8*2 10*2 8*3 10*3 16*2
%e A342085 2*2*3 2*2*4 5*2*2 12*2 15*2 4*2*4
%e A342085 3*2*2 4*2*2 3*2*4 3*2*5 4*4*2
%e A342085 2*2*2*2 4*2*3 5*2*3 8*2*2
%e A342085 4*3*2 5*3*2 2*2*2*4
%e A342085 6*2*2 2*2*4*2
%e A342085 2*2*2*3 4*2*2*2
%e A342085 2*2*3*2 2*2*2*2*2
%e A342085 3*2*2*2
%p A342085 a:= proc(n) option remember; 1+add(`if`(d>=n/d,
%p A342085 a(d), 0), d=numtheory[divisors](n) minus {n})
%p A342085 end:
%p A342085 seq(a(n), n=1..128); # _Alois P. Heinz_, Jun 24 2021
%t A342085 cmo[n_]:=Prepend[Prepend[#,n]&/@Join@@cmo/@Select[Most[Divisors[n]],#>=n/#&],{n}];
%t A342085 Table[Length[cmo[n]],{n,100}]
%Y A342085 The restriction to powers of 2 is A045690.
%Y A342085 The inferior version is A337135.
%Y A342085 The strictly inferior version is A342083.
%Y A342085 The strictly superior version is A342084.
%Y A342085 The additive version is A342094, with strict case A342095.
%Y A342085 The additive version not allowing equality is A342098.
%Y A342085 A001055 counts factorizations.
%Y A342085 A003238 counts divisibility chains summing to n-1, with strict case A122651.
%Y A342085 A038548 counts inferior (or superior) divisors.
%Y A342085 A056924 counts strictly inferior (or strictly superior) divisors.
%Y A342085 A067824 counts strict chains of divisors starting with n.
%Y A342085 A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
%Y A342085 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342085 A207375 lists central divisors.
%Y A342085 A253249 counts strict chains of divisors.
%Y A342085 A334996 counts ordered factorizations by product and length.
%Y A342085 A334997 counts chains of divisors of n by length.
%Y A342085 - Inferior: A033676, A066839, A072499, A161906.
%Y A342085 - Superior: A033677, A070038, A161908, A341676.
%Y A342085 - Strictly Inferior: A060775, A070039, A333806, A341674.
%Y A342085 - Strictly Superior: A064052/A048098, A140271, A238535, A341673.
%Y A342085 Cf. A000203, A001248, A005117, A006530, A020639, A057567, A057568, A112798, A169594, A337105, A342096, A342097.
%K A342085 nonn
%O A342085 1,4
%A A342085 _Gus Wiseman_, Feb 28 2021