A361869 Let x_0, x_1, x_2, ... be the iterations of the arithmetic derivative A003415 starting with x_0 = n. a(n) is the greatest k such that x_0 > x_1 > ... > x_k.

0, 1, 2, 2, 0, 2, 3, 2, 0, 4, 3, 2, 0, 2, 5, 1, 0, 2, 0, 2, 0, 4, 3, 2, 0, 4, 2, 0, 0, 2, 0, 2, 0, 6, 3, 1, 0, 2, 5, 1, 0, 2, 3, 2, 0, 2, 5, 2, 0, 6, 3, 1, 0, 2, 0, 1, 0, 4, 3, 2, 0, 2, 7, 2, 0, 1, 3, 2, 0, 3, 3, 2, 0, 2, 2, 2, 0, 1, 3, 2, 0, 0, 3, 2, 0, 4, 3, 1, 0, 2, 0, 1, 0, 4, 7, 1, 0, 2, 2, 3

Offset: 0

Author

Robert Israel, May 28 2023

Keywords

• nonn

Comments

a(n) is the number of iterations of A003415 starting at n until the sequence of iterates stops decreasing.

a(n) = 0 if and only if A003415(n) >= n.

First differs from A099307 at n=15, where a(15) = 1 while A099307(15) = 0.

Examples

			a(5) = 2 because x_0 = 5 > x_1 = A003415(5) = 1 > x_2 = A003415(1) = 0, but  x_3 = A003415(0) = 0.
a(6) = 3 because x_0 = 6 > x_1 = A003415(6) = 5 > ... > x_3 = 0 but x_4 = 0.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A003415, A099307.

Programs

Maple

ader:= proc(n) local t;
  n * add(t[2]/t[1], t = ifactors(n)[2])
end proc:
f:= proc(n) option remember; local t;
   t:= ader(n);
   if t < n then procname(t)+1 else 0 fi
end proc:
map(f, [$0..1000]);