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%I #17 Nov 04 2021 06:26:01
%S 1,1,1,1,1,2,1,2,1,2,1,3,1,2,2,2,1,3,1,3,2,2,1,4,1,2,2,3,1,4,1,3,2,2,
%T 2,4,1,2,2,4,1,5,1,3,3,2,1,6,1,3,2,3,1,5,2,4,2,2,1,7,1,2,3,3,2,5,1,3,
%U 2,4,1,8,1,2,3,3,2,5,1,6,2,2,1,7,2,2,2
%N Number of chains of strictly inferior divisors from n to 1.
%C We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.
%C 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 = 60/6/2/1 has first-quotients (10,3,2), which are > (6,2,1), so q is counted under a(60).
%C Also the number of factorizations of n where each factor is greater than the product of all previous factors.
%F G.f.: x + Sum_{k>=1} a(k) * x^(k*(k + 1)) / (1 - x^k). - _Ilya Gutkovskiy_, Nov 03 2021
%e The a(n) chains for n = 2, 6, 12, 24, 42, 48, 60, 72:
%e 2/1 6/1 12/1 24/1 42/1 48/1 60/1 72/1
%e 6/2/1 12/2/1 24/2/1 42/2/1 48/2/1 60/2/1 72/2/1
%e 12/3/1 24/3/1 42/3/1 48/3/1 60/3/1 72/3/1
%e 24/4/1 42/6/1 48/4/1 60/4/1 72/4/1
%e 42/6/2/1 48/6/1 60/5/1 72/6/1
%e 48/6/2/1 60/6/1 72/8/1
%e 60/6/2/1 72/6/2/1
%e 72/8/2/1
%e The a(n) factorizations for n = 2, 6, 12, 24, 42, 48, 60, 72:
%e 2 6 12 24 42 48 60 72
%e 2*3 2*6 3*8 6*7 6*8 2*30 8*9
%e 3*4 4*6 2*21 2*24 3*20 2*36
%e 2*12 3*14 3*16 4*15 3*24
%e 2*3*7 4*12 5*12 4*18
%e 2*3*8 6*10 6*12
%e 2*3*10 2*4*9
%e 2*3*12
%t cen[n_]:=If[n==1,{{1}},Prepend[#,n]&/@Join@@cen/@Select[Divisors[n],#<n/#&]];
%t Table[Length[cen[n]],{n,100}]
%Y The restriction to powers of 2 is A040039.
%Y Not requiring strict inferiority gives A074206 (ordered factorizations).
%Y The weakly inferior version is A337135.
%Y The strictly superior version is A342084.
%Y The weakly superior version is A342085.
%Y The additive version is A342098, or A000929 allowing equality.
%Y A000005 counts divisors.
%Y A001055 counts factorizations.
%Y A003238 counts chains of divisors 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 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 A342086 counts chains of divisors with strictly increasing quotients > 1.
%Y - Inferior: A033676, A066839, A072499, A161906.
%Y - Superior: A033677, A070038, A161908.
%Y - Strictly Inferior: A060775, A070039, A333806, A341674.
%Y - Strictly Superior: A048098, A064052, A140271, A238535, A341673.
%Y Cf. A000203, A001248, A002033, A006530, A018819, A020639, A045690, A337105, A342087, A342094, A342095, A342096, A342097.
%K nonn
%O 1,6
%A _Gus Wiseman_, Feb 28 2021
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