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%I #22 Jun 24 2021 20:20:53
%S 1,1,1,2,1,2,1,2,2,2,1,3,1,2,2,4,1,3,1,4,2,2,1,5,2,2,2,4,1,4,1,4,2,2,
%T 2,7,1,2,2,5,1,5,1,4,3,2,1,7,2,3,2,4,1,5,2,5,2,2,1,8,1,2,3,6,2,5,1,4,
%U 2,4,1,9,1,2,3,4,2,5,1,7,4,2,1,8,2,2,2,6,1,8,2,4,2,2,2
%N a(1) = 1; for n > 1, a(n) = Sum_{d|n, d <= sqrt(n)} a(d).
%C From _Gus Wiseman_, Mar 05 2021: (Start)
%C This sequence counts all of the following essentially equivalent things:
%C 1. Chains of distinct inferior divisors from n to 1, where a divisor d|n is inferior if d <= n/d. Inferior divisors are counted by A038548 and listed by A161906.
%C 2. Chains of divisors from n to 1 whose first-quotients (in analogy with first-differences) are term-wise greater than or equal to their decapitation (maximum element removed). For example, the divisor chain q = 60/4/2/1 has first-quotients (15,2,2), which are >= (4,2,1), so q is counted under a(60).
%C 3. Chains of divisors from n to 1 such that x >= y^2 for all adjacent x, y.
%C 4. Factorizations of n where each factor is greater than or equal to the product of all previous factors.
%C (End)
%H Alois P. Heinz, <a href="/A337135/b337135.txt">Table of n, a(n) for n = 1..65536</a>
%F G.f.: Sum_{k>=1} a(k) * x^(k^2) / (1 - x^k).
%F a(2^n) = A018819(n). - _Gus Wiseman_, Mar 08 2021
%e From _Gus Wiseman_, Mar 05 2021: (Start)
%e The a(n) chains for n = 1, 2, 4, 12, 16, 24, 36, 60:
%e 1 2/1 4/1 12/1 16/1 24/1 36/1 60/1
%e 4/2/1 12/2/1 16/2/1 24/2/1 36/2/1 60/2/1
%e 12/3/1 16/4/1 24/3/1 36/3/1 60/3/1
%e 16/4/2/1 24/4/1 36/4/1 60/4/1
%e 24/4/2/1 36/6/1 60/5/1
%e 36/4/2/1 60/6/1
%e 36/6/2/1 60/4/2/1
%e 60/6/2/1
%e The a(n) factorizations for n = 2, 4, 12, 16, 24, 36, 60:
%e 2 4 12 16 24 36 60
%e 2*2 2*6 2*8 3*8 4*9 2*30
%e 3*4 4*4 4*6 6*6 3*20
%e 2*2*4 2*12 2*18 4*15
%e 2*2*6 3*12 5*12
%e 2*2*9 6*10
%e 2*3*6 2*2*15
%e 2*3*10
%e (End)
%p a:= proc(n) option remember; `if`(n=1, 1, add(
%p `if`(d<=n/d, a(d), 0), d=numtheory[divisors](n)))
%p end:
%p seq(a(n), n=1..128); # _Alois P. Heinz_, Jun 24 2021
%t a[1] = 1; a[n_] := a[n] = DivisorSum[n, a[#] &, # <= Sqrt[n] &]; Table[a[n], {n, 95}]
%t (* second program *)
%t asc[n_]:=Prepend[#,n]&/@Prepend[Join@@Table[asc[d],{d,Select[Divisors[n],#<n&&#<=n/#&]}],{}];Table[Length[Select[asc[n],MemberQ[#,1]&]],{n,100}] (* _Gus Wiseman_, Mar 05 2021 *)
%Y Cf. A002033, A008578 (positions of 1's), A068108.
%Y The restriction to powers of 2 is A018819.
%Y Not requiring inferiority gives A074206 (ordered factorizations).
%Y The strictly inferior version is A342083.
%Y The strictly superior version is A342084.
%Y The weakly superior version is A342085.
%Y The additive version is A000929, or A342098 forbidding equality.
%Y A000005 counts divisors, with sum A000203.
%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 strict factorizations of divisors.
%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. A001248, A006530, A020639, A040039, A045690, A337105, A342087, A342094, A342095, A342096, A342097.
%K nonn
%O 1,4
%A _Ilya Gutkovskiy_, Nov 21 2020
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