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$\frac n2$ is an integer implies n is even. The smallest four-digit base 9 integer is $(1000)_9$, which is $9^3 = 729$, but this is not even, so $\frac n2$ is at least 729 + 1 = 730. This gives the lower bound $n \geq 1460$.

While n/2 is a four-digit base 9 integer, 3n cannot exceed the maximum value of four-digit base 9 integers. that value is $(8888)_9$, which is $8(9^3 + 9^2 + 9 + 1) = 6560$, but it is not divisible by 3. That means $3n \leq 6558$, i.e., $n \leq 2186$ is the upper bound.

To ensure that n/2 is an integer, the answer is the number of even numbers n in the range $1460 \leq n \leq 2186$, i.e., $\dfrac{2186 - 1460}2 + 1 = \boxed{364}$.