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 A342084 #21 Nov 02 2024 09:13:56
%S A342084 1,1,1,1,1,2,1,2,1,2,1,4,1,2,2,3,1,4,1,4,2,2,1,9,1,2,2,4,1,7,1,6,2,2,
%T A342084 2,10,1,2,2,9,1,6,1,4,4,2,1,19,1,4,2,4,1,8,2,9,2,2,1,20,1,2,4,10,2,6,
%U A342084 1,4,2,7,1,29,1,2,4,4,2,6,1,19,3,2,1,19,2
%N A342084 Number of chains of distinct strictly superior divisors starting with n.
%C A342084 We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.
%C A342084 These chains have first-quotients (in analogy with first-differences) that are term-wise < their decapitation (maximum element removed). Equivalently, x < y^2 for all adjacent x, y. For example, the divisor chain q = 30/6/3 has first-quotients (5,2), which are < (6,3), so q is counted under a(30).
%C A342084 Also the number of ordered factorizations of n where each factor is less than the product of all previous factors.
%H A342084 Amiram Eldar, <a href="/A342084/b342084.txt">Table of n, a(n) for n = 1..10000</a>
%F A342084 a(2^n) = A045690(n).
%e A342084 The a(n) chains for n = 2, 6, 12, 16, 24, 30, 32, 36:
%e A342084 2 6 12 16 24 30 32 36
%e A342084 6/3 12/4 16/8 24/6 30/6 32/8 36/9
%e A342084 12/6 16/8/4 24/8 30/10 32/16 36/12
%e A342084 12/6/3 24/12 30/15 32/8/4 36/18
%e A342084 24/6/3 30/6/3 32/16/8 36/12/4
%e A342084 24/8/4 30/10/5 32/16/8/4 36/12/6
%e A342084 24/12/4 30/15/5 36/18/6
%e A342084 24/12/6 36/18/9
%e A342084 24/12/6/3 36/12/6/3
%e A342084 36/18/6/3
%e A342084 The a(n) ordered factorizations for n = 2, 6, 12, 16, 24, 30, 32, 36:
%e A342084 2 6 12 16 24 30 32 36
%e A342084 3*2 4*3 8*2 6*4 6*5 8*4 9*4
%e A342084 6*2 4*2*2 8*3 10*3 16*2 12*3
%e A342084 3*2*2 12*2 15*2 4*2*4 18*2
%e A342084 3*2*4 3*2*5 8*2*2 4*3*3
%e A342084 4*2*3 5*2*3 4*2*2*2 6*2*3
%e A342084 4*3*2 5*3*2 6*3*2
%e A342084 6*2*2 9*2*2
%e A342084 3*2*2*2 3*2*2*3
%e A342084 3*2*3*2
%t A342084 ceo[n_]:=Prepend[Prepend[#,n]&/@Join@@ceo/@Select[Most[Divisors[n]],#>n/#&],{n}];
%t A342084 Table[Length[ceo[n]],{n,100}]
%Y A342084 The restriction to powers of 2 is A045690, with reciprocal version A040039.
%Y A342084 The inferior version is A337135.
%Y A342084 The strictly inferior version is A342083.
%Y A342084 The superior version is A342085.
%Y A342084 The additive version allowing equality is A342094 or A342095.
%Y A342084 The additive version is A342096 or A342097.
%Y A342084 A000005 counts divisors.
%Y A342084 A001055 counts factorizations.
%Y A342084 A003238 counts divisibility chains summing to n-1, with strict case A122651.
%Y A342084 A038548 counts inferior (or superior) divisors.
%Y A342084 A056924 counts strictly inferior (or strictly superior) divisors.
%Y A342084 A067824 counts strict chains of divisors starting with n.
%Y A342084 A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
%Y A342084 A167865 counts strict chains of divisors > 1 summing to n.
%Y A342084 A207375 lists central divisors.
%Y A342084 A253249 counts strict chains of divisors.
%Y A342084 A334996 counts ordered factorizations by product and length.
%Y A342084 A334997 counts chains of divisors of n by length.
%Y A342084 - Inferior: A033676, A063962, A066839, A072499, A161906.
%Y A342084 - Superior: A033677, A070038, A161908, A341591.
%Y A342084 - Strictly Inferior: A060775, A070039, A333806, A341674.
%Y A342084 - Strictly Superior: A064052/A048098, A140271, A238535, A341642, A341673.
%Y A342084 Cf. A000203, A000929, A001248, A006530, A018819, A020639, A169594, A337105, A342098.
%K A342084 nonn
%O A342084 1,6
%A A342084 _Gus Wiseman_, Feb 28 2021