10. 11 .

Let $\left(a_{n} a_{n,-1}, \cdots a_{n}\right):=a_{n} \times 7^{n}+a_{n}: \times 7^{n-1}+\cdots+a_{n}$ be a good number, where $a_{2} \in\{0,1, \cdots, 6\}$, $a_{n} \neq 0$. From the condition, we have
$$
a_{n} \times 10^{n}+a_{n} \times 10^{n-1}+\cdots-a_{0}-2\left(a_{n} \times 7^{\pi}+a_{n} \times 7^{\pi 1}+\cdots+a_{0}\right)
$$
$14 b+2 c$, i.e., $c=2(a-2 b)$, so $0 \leqslant b \leqslant 3$. By discussing each case, we find that all the good numbers are $(102)_{7} ，(204)_{7}$, $(306)_{\uparrow},(210)_{7},(312)_{7},(414)_{7},(516)_{7},(420)_{7},(522)_{i},(624)_{7},(630)$, a total of 11.
If a positive integer, when its base-7 representation is read as a decimal, results in a number that is twice the original number, then it is called a "good number". Then the number of "good numbers" is $\qquad$.

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