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A367632
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Numbers k with at least one pair of adjacent divisors i,j of k with i < j such that i' > j'. Here, n' denotes the arithmetic derivative of n.
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0
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20, 28, 40, 42, 44, 52, 56, 60, 66, 68, 72, 76, 78, 80, 84, 88, 92, 99, 100, 102, 104, 110, 112, 114, 116, 117, 120, 124, 126, 130, 132, 136, 138, 140, 144, 148, 152, 153, 156, 160, 164, 168, 170, 171, 172, 174, 176, 180, 184, 186, 188, 190, 196, 198, 200, 204
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OFFSET
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1,1
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COMMENTS
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Numbers whose list of arithmetic derivatives of its divisors has at least one descent (in terms of its increasing list of divisors).
First departs from A137428 at a(11) = 72.
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LINKS
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EXAMPLE
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72 is in the sequence. The divisors of 72 are {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72} and their corresponding derivatives are {0, 1, 1, 4, 5, 12, 6, 16, 21, 44, 60, 156}. Since 8 < 9, and 8' = 12 > 6 = 9', 72 is a member.
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MATHEMATICA
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ad[1] = 0; ad[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); q[n_] := Module[{d = Divisors[n]}, AnyTrue[Range[Length[d] - 1], ad[d[[#]]] > ad[d[[# + 1]]] &]]; Select[Range[200], q] (* Amiram Eldar, Nov 25 2023 *)
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CROSSREFS
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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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