imc

Looking at the first few terms of the sequence: $a_1=2,a_2=3,a_3=\frac{3}{2},a_4=\frac{1}{2},a_5=\frac{1}{3},a_6=\frac{2}{3},a_7=2,a_8=3,....$ Clearly, the sequence repeats every 6 terms. Since $2006\equiv 2\mathrm{mod}6$, $a_{2006}=a_2=\boxed{3}$