recursion - Solving problem using recursive functions

Attached below is a question posed by the Canadian Mathematical Society, and I have my code and answer. Is there a better way of writing the code, and will the answer be different as a result?

My code and possible answer (edited):

jk[0] = 0; 
jk[1] = 0; 
jk[2] = 1; 

jk[3] = 2; 
tr[n_] := If[n > 3, 
  LengthWhile[Range[3, 20], Divisible[n, #1] & ] + 3, jk[n]]
Sum[tr[tr[tr[m]]], {m, 1, 2006}]

(* 2672 *)

Let me know if you get an answer that differs.

Answer

You made a few mistakes. jk[0] should be 0 in your code and your function tr is wrong.

Corrected version:

t[0] = 0;
t[1] = 0;
t[2] = 1;
t[n_] := t[n] = LengthWhile[Range[1, 11], Divisible[n, #1] &] + 1
Sum[Nest[t, m, 3], {m, 1, 2006}]

1171

10x faster version:

t[n_] := t[n] = Module[{i = 1}, While[MemberQ[Divisors[n], i], i++]; i];