II. (50 points) Define a "Hope Set" (Hope Set) abbreviated as HS as follows: HS is a non-empty set that satisfies the condition "if $x \in \mathrm{HS}$, then $2 x \notin \mathrm{HS}$". How many "Hope Subsets" are there in the set $\{1,2, \cdots, 30\}$? Please explain your reasoning.
Sure, here is the translation:

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II. Below, “ $a \longmapsto 2 a$ ” represents the doubling relationship between $a$ and $2 a$. Notice:
$\left\{\begin{array}{ll}\text { (1) } & 15 \longmapsto 30, \\ \text { (2) } & 13 \longmapsto 26, \\ \text { (3) } & 11 \longmapsto 22, \\ \text { (4) } & 9 \longmapsto 18,\end{array}\right.$
$\left\{\begin{array}{ll}(5) & 7 \longmapsto 14,14 \longmapsto 28, \\ \text { (6) } & 5 \longmapsto 10,10 \longmapsto 20,\end{array}\right.$
(7) $3 \longmapsto 6,6 \longmapsto 12,12 \longrightarrow 24$,
(8) $1 \longmapsto 2,2 \longmapsto 4,4 \longmapsto 8,8 \longmapsto 16$.
Clearly, $17,19,21,23,25,27,29$ do not affect whether HS is a hopeful subset (since these numbers are not divisible by 2, and each number's double is greater than 30), so there are $2^{7}$ ways to assign these 7 numbers.

In (1), 15 and 30 cannot both be taken, so there are $2^{2}-1=3$ ways.
Similarly, in (2), (3), and (4), there are 3 ways each.
Next, we use a recursive algorithm.
In (5), if 7 is taken, then 14 cannot be taken, and 28 can be taken or not, giving 2 ways; if 7 is not taken, then by (1), there are 3 ways for 14 and 28 (the situation for 14 and 28 is the same as in (1)). Therefore, there are $2+3=5$ ways in (5).
Similarly, there are 5 ways in (6).
In (7), if 3 is taken, then 6 cannot be taken, and by (1), there are 3 ways for 12 and 24; if 3 is not taken, then by (5), there are 5 ways for $6,12,24$. Therefore, there are $3+5=8$ ways in (7).

In (8), if 1 is taken, then 2 cannot be taken, and by (5), there are 5 ways for 4,8,16; if 1 is not taken, then by (7), there are 8 ways for $2,4,8,16$. Therefore, there are $5+8=13$ ways in (8).

Considering the empty set $\varnothing$ (i.e., none of $1,2, \cdots, 30$ are taken), the number of hopeful subsets of $\{1,2, \cdots, 30\}$ is
$$
2^{7} \times 3^{4} \times 5^{2} \times 8 \times 13-1=26956799 \text {. }
$$
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