This is from Buell, Binary Quadratic Forms, pages 109-119.

Note that $h(-16) = h(-4) = 1.$

When $p$ is an odd prime, we use the Legndre symbol in
$$ h(-4p^2) = \frac{p - (-1|p)}{2} $$

After that, if $u > 1,$ while $u$ is allowed odd or even as needed; if $p$ does not divide $u,$ then
$$ h(-4u^2p^2) = h(-4 u^2) \cdot \left(p - (-1|p) \right). $$
If $p$ divides $u,$
$$ h(-4u^2p^2) = h(-4 u^2) \cdot p. $$
$$ h(-16u^2) = 2 h(-4 u^2) $$

Let's see; note that $p - (-1|p)$ for an odd prime $p$ is always divisble by $4,$ so these class numbers are even, and the max power of $2$ grows. This is fair, the number of genera is growing with each new prime and is always a power of two, while each discriminant has a constant number of classes per genus.

For example, with odd $n > 1,$ if $$ n = \prod_p p^{e_p} $$ we get
$$ h(-4n^2) = \frac{1}{2} \prod_p \left( p - (-1|p)\right) p^{e_p -1} $$

The conclusion is that $h(-4n^2)$ is mostly pretty similar in size to $\frac{n}{2} .$ In order to deviate much from that, it is most efficient to have $n$ squarefree odd, with either $n = 5 \cdot 13 \cdot 17 ...$ the product of the consecutive primes $1 \pmod 4$ up to some bound $P,$ or $n = 3 \cdot 7 \cdot 11 \cdot 19 ...$ the product of the consecutive primes $3 \pmod 4$ up to some bound. In these cases, the growth or shrinkage of $h$ is still bounded by Merten's Theorem, in one version (H+W_thm427)
$\sum_{p \leq x} \frac{1}{p} = \log \log x + B + o(1).$ I guess if we are taking half the primes according to mod 4, we would expect the estimate to be halved.

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```
n -4 n^2 h n factored
1 -4 1 factor n = 1
2 -16 1 factor n = 2
3 -36 2 factor n = 3
4 -64 2 factor n = 2^2
5 -100 2 factor n = 5
6 -144 4 factor n = 2 * 3
7 -196 4 factor n = 7
8 -256 4 factor n = 2^3
9 -324 6 factor n = 3^2
10 -400 4 factor n = 2 * 5
11 -484 6 factor n = 11
12 -576 8 factor n = 2^2 * 3
13 -676 6 factor n = 13
14 -784 8 factor n = 2 * 7
15 -900 8 factor n = 3 * 5
16 -1024 8 factor n = 2^4
17 -1156 8 factor n = 17
18 -1296 12 factor n = 2 * 3^2
19 -1444 10 factor n = 19
20 -1600 8 factor n = 2^2 * 5
21 -1764 16 factor n = 3 * 7
22 -1936 12 factor n = 2 * 11
23 -2116 12 factor n = 23
24 -2304 16 factor n = 2^3 * 3
25 -2500 10 factor n = 5^2
26 -2704 12 factor n = 2 * 13
27 -2916 18 factor n = 3^3
28 -3136 16 factor n = 2^2 * 7
29 -3364 14 factor n = 29
30 -3600 16 factor n = 2 * 3 * 5
31 -3844 16 factor n = 31
32 -4096 16 factor n = 2^5
33 -4356 24 factor n = 3 * 11
34 -4624 16 factor n = 2 * 17
35 -4900 16 factor n = 5 * 7
36 -5184 24 factor n = 2^2 * 3^2
37 -5476 18 factor n = 37
38 -5776 20 factor n = 2 * 19
39 -6084 24 factor n = 3 * 13
40 -6400 16 factor n = 2^3 * 5
41 -6724 20 factor n = 41
42 -7056 32 factor n = 2 * 3 * 7
43 -7396 22 factor n = 43
44 -7744 24 factor n = 2^2 * 11
45 -8100 24 factor n = 3^2 * 5
46 -8464 24 factor n = 2 * 23
47 -8836 24 factor n = 47
48 -9216 32 factor n = 2^4 * 3
49 -9604 28 factor n = 7^2
50 -10000 20 factor n = 2 * 5^2
51 -10404 32 factor n = 3 * 17
52 -10816 24 factor n = 2^2 * 13
53 -11236 26 factor n = 53
54 -11664 36 factor n = 2 * 3^3
55 -12100 24 factor n = 5 * 11
56 -12544 32 factor n = 2^3 * 7
57 -12996 40 factor n = 3 * 19
58 -13456 28 factor n = 2 * 29
59 -13924 30 factor n = 59
60 -14400 32 factor n = 2^2 * 3 * 5
61 -14884 30 factor n = 61
62 -15376 32 factor n = 2 * 31
63 -15876 48 factor n = 3^2 * 7
64 -16384 32 factor n = 2^6
65 -16900 24 factor n = 5 * 13
66 -17424 48 factor n = 2 * 3 * 11
67 -17956 34 factor n = 67
68 -18496 32 factor n = 2^2 * 17
69 -19044 48 factor n = 3 * 23
70 -19600 32 factor n = 2 * 5 * 7
71 -20164 36 factor n = 71
72 -20736 48 factor n = 2^3 * 3^2
73 -21316 36 factor n = 73
74 -21904 36 factor n = 2 * 37
75 -22500 40 factor n = 3 * 5^2
76 -23104 40 factor n = 2^2 * 19
77 -23716 48 factor n = 7 * 11
78 -24336 48 factor n = 2 * 3 * 13
79 -24964 40 factor n = 79
80 -25600 32 factor n = 2^4 * 5
81 -26244 54 factor n = 3^4
82 -26896 40 factor n = 2 * 41
83 -27556 42 factor n = 83
84 -28224 64 factor n = 2^2 * 3 * 7
85 -28900 32 factor n = 5 * 17
86 -29584 44 factor n = 2 * 43
87 -30276 56 factor n = 3 * 29
88 -30976 48 factor n = 2^3 * 11
89 -31684 44 factor n = 89
90 -32400 48 factor n = 2 * 3^2 * 5
91 -33124 48 factor n = 7 * 13
92 -33856 48 factor n = 2^2 * 23
93 -34596 64 factor n = 3 * 31
94 -35344 48 factor n = 2 * 47
95 -36100 40 factor n = 5 * 19
96 -36864 64 factor n = 2^5 * 3
97 -37636 48 factor n = 97
98 -38416 56 factor n = 2 * 7^2
99 -39204 72 factor n = 3^2 * 11
100 -40000 40 factor n = 2^2 * 5^2
n -4 n^2 h n factored
```

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