jmc

If $a_1$ is even, then $a_2=\frac{a_1}{2}<a_1,$ so $a_1$ does not have the given property.

If $a_1$ is of the form $4k+1,$ then $a_2=3(4k+1)+1=12k+4,$ $a_3=6k+2,$ and $$a_4=3k+1<a_1,$$so $a_1$ does not have the given property in this case either.

If $a_1$ is of the form $4k+3,$ then $a_2=3(4k+3)+1=12k+10,$ $a_3=6k+5,$ and $$a_4=3(6k+5)+1=18k+16,$$which are all greater than $a_1,$ so in this case, $a_1$ has the given property.

There are $2008/4=502$ numbers less than or equal to 2008 that have the form $4k+3.$ Thus, the answer is $\boxed{502}.$