3. Find the smallest positive integer that can be expressed as the sum of the squares of four positive integers and can divide some integer of the form $2^{n}+15\left(n \in \mathbf{N}_{+}\right)$.
3. The smallest five positive integers that can be expressed as the sum of four positive integer squares are
$$
\begin{array}{l}
4=1+1+1+1, \\
7=4+1+1+1, \\
10=4+4+1+1, \\
12=9+1+1+1, \\
13=4+4+4+1 .
\end{array}
$$

Obviously, since $2^{n}+15$ is odd, the smallest positive integer cannot be 4, 10, or 12.

Also, $2^{n} \equiv 1,2,4 \neq-1(\bmod 7)$, so $2^{n}+15$ is not a multiple of 7.

And $2^{7}+15=143=11 \times 13$, hence the smallest positive integer that meets the condition is 13.