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.

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

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

Table of n, a(n) for n=1..56.

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

Cf. A003415 (n'), A137428.

Sequence in context: A269986 A139805 A141823 * A137428 A368089 A367455

Adjacent sequences: A367629 A367630 A367631 * A367633 A367634 A367635

Keyword

nonn

Author

Wesley Ivan Hurt, Nov 24 2023

Status

approved