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%I #16 Nov 26 2023 10:52:39
%S 20,28,40,42,44,52,56,60,66,68,72,76,78,80,84,88,92,99,100,102,104,
%T 110,112,114,116,117,120,124,126,130,132,136,138,140,144,148,152,153,
%U 156,160,164,168,170,171,172,174,176,180,184,186,188,190,196,198,200,204
%N 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.
%C Numbers whose list of arithmetic derivatives of its divisors has at least one descent (in terms of its increasing list of divisors).
%C First departs from A137428 at a(11) = 72.
%e 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.
%t 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 *)
%Y Cf. A003415 (n'), A137428.
%K nonn
%O 1,1
%A _Wesley Ivan Hurt_, Nov 24 2023