4. Let $n$ be a natural number. If 2005 can be written as the sum of $n$ positive odd composite numbers, then $n$ is called a "good number". The number of such good numbers is $\qquad$.

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4.111.

Let $a_{1}, a_{2}, \cdots, a_{n}$ be odd composite numbers, and $a_{1}+a_{2}+\cdots+a_{n}=$ 2005, then $n$ is odd. Since 9 is the smallest odd composite number, and 2005 $<2007=9 \times 223$, hence $n<223$. Therefore, $n \leqslant 221$. 
$$
2005=1980+25=\underbrace{9+9+\cdots}_{20 \uparrow}+9+25 \text {, }
$$

Thus, 221 is a good number.
Furthermore, when $(2 k-1) \times 9$ is an odd composite number,
$$
(2 k-1) \times 9+9+9=(2 k+1) \times 9
$$

is also an odd composite number. Therefore, the terms on the right side of equation (1) can be combined step by step, so $n$ can take 221, 219, ..., 5, 3.
Since 2005 itself is also an odd composite number, then $n$ can take 1.
Thus, 1, 3, 5, ..., 221 are all good numbers, totaling 111.