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A363373 a(n) is the least k such that, if x_0, x_1, x_2, ... are the iterations of the arithmetic derivative A003415 starting with x_0 = k, x_0 > x_1 > ... > x_n.

%I A363373 #6 May 30 2023 07:45:57
%S A363373 0,1,2,6,9,14,33,62,177,886,1155,1719,3255,4018,13377,19942,46022,
%T A363373 103401,193426,422751,634113,1080742,2850591,5493662,10252635,
%U A363373 25631525,51217666,135055839
%N A363373 a(n) is the least k such that, if x_0, x_1, x_2, ... are the iterations of the arithmetic derivative A003415 starting with x_0 = k, x_0 > x_1 > ... > x_n.
%C A363373 a(n) is the least k such that the first n iterations of A003415 starting at k are decreasing.
%C A363373 a(n) is the least k such that A361869(k) = n.
%e A363373 a(3) = 6 because the iterations of A003415 starting at 6 are 6 > 5 > 1 > 0 = 0.
%e A363373 First differs from A189760 and A327967 at 9, where a(9) = 886 (corresponding to iterations 886 > 445 > 94 > 49 > 14 > 9 > 6 > 5 > 1 > 0) while A189760(9) = A327967(9) = 414 < A003415(414) = 501.
%p A363373 ader:= proc(n) local t;
%p A363373   n * add(t[2]/t[1], t = ifactors(n)[2])
%p A363373 end proc:
%p A363373 f:= proc(n) option remember; local t;
%p A363373    t:= ader(n);
%p A363373    if t < n then procname(t)+1 else 0 fi
%p A363373 end proc:
%p A363373 M:= 25: V:= Array(0..M,-1): count:= 0:
%p A363373 for n from 0 while count <= M do
%p A363373   v:= f(n);
%p A363373   if V[v] = -1 then count:= count+1; V[v]:= n fi;
%p A363373 od:
%p A363373 convert(V,list);
%Y A363373 Cf. A003415, A189760, A327967, A361869.
%K A363373 nonn,more
%O A363373 0,3
%A A363373 _Robert Israel_, May 29 2023