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A342085
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Number of decreasing chains of distinct superior divisors starting with n.
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12
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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, 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, 2, 6, 1, 4, 2, 7, 1, 39, 1, 2, 5, 4, 2, 6, 1, 23, 6, 2, 1
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OFFSET
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1,4
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COMMENTS
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We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.
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).
Also the number of ordered factorizations of n where each factor is less than or equal to the product of all previous factors.
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LINKS
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FORMULA
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EXAMPLE
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The a(n) chains for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
2 4 8 12 16 20 24 30 32
4/2 8/4 12/4 16/4 20/5 24/6 30/6 32/8
8/4/2 12/6 16/8 20/10 24/8 30/10 32/16
12/4/2 16/4/2 20/10/5 24/12 30/15 32/8/4
12/6/3 16/8/4 24/6/3 30/6/3 32/16/4
16/8/4/2 24/8/4 30/10/5 32/16/8
24/12/4 30/15/5 32/8/4/2
24/12/6 32/16/4/2
24/8/4/2 32/16/8/4
24/12/4/2 32/16/8/4/2
24/12/6/3
The a(n) ordered factorizations for n = 2, 4, 8, 12, 16, 20, 24, 30, 32:
2 4 8 12 16 20 24 30 32
2*2 4*2 4*3 4*4 5*4 6*4 6*5 8*4
2*2*2 6*2 8*2 10*2 8*3 10*3 16*2
2*2*3 2*2*4 5*2*2 12*2 15*2 4*2*4
3*2*2 4*2*2 3*2*4 3*2*5 4*4*2
2*2*2*2 4*2*3 5*2*3 8*2*2
4*3*2 5*3*2 2*2*2*4
6*2*2 2*2*4*2
2*2*2*3 4*2*2*2
2*2*3*2 2*2*2*2*2
3*2*2*2
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MAPLE
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a:= proc(n) option remember; 1+add(`if`(d>=n/d,
a(d), 0), d=numtheory[divisors](n) minus {n})
end:
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MATHEMATICA
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cmo[n_]:=Prepend[Prepend[#, n]&/@Join@@cmo/@Select[Most[Divisors[n]], #>=n/#&], {n}];
Table[Length[cmo[n]], {n, 100}]
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CROSSREFS
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The restriction to powers of 2 is A045690.
The strictly inferior version is A342083.
The strictly superior version is A342084.
The additive version not allowing equality is A342098.
A003238 counts divisibility chains summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
A167865 counts strict chains of divisors > 1 summing to n.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
Cf. A000203, A001248, A005117, A006530, A020639, A057567, A057568, A112798, A169594, A337105, A342096, A342097.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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