

A367632


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.


0



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

1,1


COMMENTS

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.


LINKS



EXAMPLE

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.


MATHEMATICA

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 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



