dwavesystems / dwavesystems/dimod

make_quadratic problem: failed to produce a correct result.

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Description

https://github.com/dwavesystems/dimod/blob/434f75d0fc958a6bc3e2658ec9bc758346a39f00/dimod/higherorder/utils.py#L102

That function does not lead to a correct sample set. Here is a simple test case (factoring).

I want to factor 15 to 3 and 5 and I know 6 variables are enough. But, I want to test the `make_quadratic `with more variables, so I use 16 variables.

```
a = 128*q7 + 64*q6 + 32*q5 + 16*q4 + 8*q3 + 4*q2 + 2*q1 + q0
b = 128*q15 + 64*q14 + 32*q13 + 16*q12 + 8*q11 + 4*q10 + 2*q9 + q8
```

Expand `P = (15 - a*b)**2`, and you will get the polynomial shown below.

Then, do this:

```
strength = 10; # or whatever your recommend
bqm = make_quadratic(poly , strength, dimod.BINARY)
```

I tried the real QPU with `DWaveSampler `of various solvers on that binary quadratic model. The results are always wrong. Could you please help address the issue? Thanks.

```
poly = {
(): 225,
(0, 1, 8): 4,
(0, 1, 8, 9): 16,
(0, 1, 9): 16,
(0, 1, 10): 64,
(0, 1, 10, 8): 32,
(0, 1, 10, 9): 64,
(0, 1, 10, 11): 256,
(0, 1, 10, 12): 512,
(0, 1, 10, 13): 1024,
(0, 1, 10, 14): 2048,
(0, 1, 10, 15): 4096,
(0, 1, 11): 256,
(0, 1, 11, 8): 64,
(0, 1, 11, 9): 128,
(0, 1, 11, 12): 1024,
(0, 1, 11, 13): 2048,
(0, 1, 11, 14): 4096,
(0, 1, 11, 15): 8192,
(0, 1, 12): 1024,
(0, 1, 12, 8): 128,
(0, 1, 12, 9): 256,
(0, 1, 12, 13): 4096,
(0, 1, 12, 14): 8192,
(0, 1, 12, 15): 16384,
(0, 1, 13): 4096,
(0, 1, 13, 8): 256,
(0, 1, 13, 9): 512,
(0, 1, 13, 14): 16384,
(0, 1, 13, 15): 32768,
(0, 1, 14): 16384,
(0, 1, 14, 8): 512,
(0, 1, 14, 9): 1024,
(0, 1, 14, 15): 65536,
(0, 1, 15): 65536,
(0, 1, 15, 8): 1024,
(0, 1, 15, 9): 2048,
(0, 2, 8): 8,
(0, 2, 8, 9): 32,
(0, 2, 9): 32,
(0, 2, 10): 128,
(0, 2, 11): 512,
(0, 2, 12): 2048,
(0, 2, 13): 8192,
(0, 2, 14): 32768,
(0, 2, 15): 131072,
(0, 3, 8): 16,
(0, 3, 8, 9): 64,
(0, 3, 9): 64,
(0, 3, 10): 256,
(0, 3, 11): 1024,
(0, 3, 12): 4096,
(0, 3, 13): 16384,
(0, 3, 14): 65536,
(0, 3, 15): 262144,
(0, 4, 8): 32,
(0, 4, 8, 9): 128,
(0, 4, 9): 128,
(0, 4, 10): 512,
(0, 4, 11): 2048,
(0, 4, 12): 8192,
(0, 4, 13): 32768,
(0, 4, 14): 131072,
(0, 4, 15): 524288,
(0, 5, 8): 64,
(0, 5, 8, 9): 256,
(0, 5, 9): 256,
(0, 5, 10): 1024,
(0, 5, 11): 4096,
(0, 5, 12): 16384,
(0, 5, 13): 65536,
(0, 5, 14): 262144,
(0, 5, 15): 1048576,
(0, 6, 8): 128,
(0, 6, 8, 9): 512,
(0, 6, 9): 512,
(0, 6, 10): 2048,
(0, 6, 11): 8192,
(0, 6, 12): 32768,
(0, 6, 13): 131072,
(0, 6, 14): 524288,
(0, 6, 15): 2097152,
(0, 7, 8): 256,
(0, 7, 8, 9): 1024,
(0, 7, 9): 1024,
(0, 7, 10): 4096,
(0, 7, 11): 16384,
(0, 7, 12): 65536,
(0, 7, 13): 262144,
(0, 7, 14): 1048576,
(0, 7, 15): 4194304,
(0, 8): -29,
(0, 9): -56,
(0, 10): -104,
(0, 10, 2, 8): 64,
(0, 10, 2, 9): 128,
(0, 10, 3, 8): 128,
(0, 10, 3, 9): 256,
(0, 10, 4, 8): 256,
(0, 10, 4, 9): 512,
(0, 10, 5, 8): 512,
(0, 10, 5, 9): 1024,
(0, 10, 6, 8): 1024,
(0, 10, 6, 9): 2048,
(0, 10, 7, 8): 2048,
(0, 10, 7, 9): 4096,
(0, 10, 11, 2): 512,
(0, 10, 11, 3): 1024,
(0, 10, 11, 4): 2048,
(0, 10, 11, 5): 4096,
(0, 10, 11, 6): 8192,
(0, 10, 11, 7): 16384,
(0, 10, 12, 2): 1024,
(0, 10, 12, 3): 2048,
(0, 10, 12, 4): 4096,
(0, 10, 12, 5): 8192,
(0, 10, 12, 6): 16384,
(0, 10, 12, 7): 32768,
(0, 10, 13, 2): 2048,
(0, 10, 13, 3): 4096,
(0, 10, 13, 4): 8192,
(0, 10, 13, 5): 16384,
(0, 10, 13, 6): 32768,
(0, 10, 13, 7): 65536,
(0, 10, 14, 2): 4096,
(0, 10, 14, 3): 8192,
(0, 10, 14, 4): 16384,
(0, 10, 14, 5): 32768,
(0, 10, 14, 6): 65536,
(0, 10, 14, 7): 131072,
(0, 10, 15, 2): 8192,
(0, 10, 15, 3): 16384,
(0, 10, 15, 4): 32768,
(0, 10, 15, 5): 65536,
(0, 10, 15, 6): 131072,
(0, 10, 15, 7): 262144,
(0, 11): -176,
(0, 11, 2, 8): 128,
(0, 11, 2, 9): 256,
(0, 11, 3, 8): 256,
(0, 11, 3, 9): 512,
(0, 11, 4, 8): 512,
(0, 11, 4, 9): 1024,
(0, 11, 5, 8): 1024,
(0, 11, 5, 9): 2048,
(0, 11, 6, 8): 2048,
(0, 11, 6, 9): 4096,
(0, 11, 7, 8): 4096,
(0, 11, 7, 9): 8192,
(0, 11, 12, 2): 2048,
(0, 11, 12, 3): 4096,
(0, 11, 12, 4): 8192,
(0, 11, 12, 5): 16384,
(0, 11, 12, 6): 32768,
(0, 11, 12, 7): 65536,
(0, 11, 13, 2): 4096,
(0, 11, 13, 3): 8192,
(0, 11, 13, 4): 16384,
(0, 11, 13, 5): 32768,
(0, 11, 13, 6): 65536,
(0, 11, 13, 7): 131072,
(0, 11, 14, 2): 8192,
(0, 11, 14, 3): 16384,
(0, 11, 14, 4): 32768,
(0, 11, 14, 5): 65536,
(0, 11, 14, 6): 131072,
(0, 11, 14, 7): 262144,
(0, 11, 15, 2): 16384,
(0, 11, 15, 3): 32768,
(0, 11, 15, 4): 65536,
(0, 11, 15, 5): 131072,
(0, 11, 15, 6): 262144,
(0, 11, 15, 7): 524288,
(0, 12): -224,
(0, 12, 2, 8): 256,
(0, 12, 2, 9): 512,
(0, 12, 3, 8): 512,
(0, 12, 3, 9): 1024,
(0, 12, 4, 8): 1024,
(0, 12, 4, 9): 2048,
(0, 12, 5, 8): 2048,
(0, 12, 5, 9): 4096,
(0, 12, 6, 8): 4096,
(0, 12, 6, 9): 8192,
(0, 12, 7, 8): 8192,
(0, 12, 7, 9): 16384,
(0, 12, 13, 2): 8192,
(0, 12, 13, 3): 16384,
(0, 12, 13, 4): 32768,
(0, 12, 13, 5): 65536,
(0, 12, 13, 6): 131072,
(0, 12, 13, 7): 262144,
(0, 12, 14, 2): 16384,
(0, 12, 14, 3): 32768,
(0, 12, 14, 4): 65536,
(0, 12, 14, 5): 131072,
(0, 12, 14, 6): 262144,
(0, 12, 14, 7): 524288,
(0, 12, 15, 2): 32768,
(0, 12, 15, 3): 65536,
(0, 12, 15, 4): 131072,
(0, 12, 15, 5): 262144,
(0, 12, 15, 6): 524288,
(0, 12, 15, 7): 1048576,
(0, 13): 64,
(0, 13, 2, 8): 512,
(0, 13, 2, 9): 1024,
(0, 13, 3, 8): 1024,
(0, 13, 3, 9): 2048,
(0, 13, 4, 8): 2048,
(0, 13, 4, 9): 4096,
(0, 13, 5, 8): 4096,
(0, 13, 5, 9): 8192,
(0, 13, 6, 8): 8192,
(0, 13, 6, 9): 16384,
(0, 13, 7, 8): 16384,
(0, 13, 7, 9): 32768,
(0, 13, 14, 2): 32768,
(0, 13, 14, 3): 65536,
(0, 13, 14, 4): 131072,
(0, 13, 14, 5): 262144,
(0, 13, 14, 6): 524288,
(0, 13, 14, 7): 1048576,
(0, 13, 15, 2): 65536,
(0, 13, 15, 3): 131072,
(0, 13, 15, 4): 262144,
(0, 13, 15, 5): 524288,
(0, 13, 15, 6): 1048576,
(0, 13, 15, 7): 2097152,
(0, 14): 2176,
(0, 14, 2, 8): 1024,
(0, 14, 2, 9): 2048,
(0, 14, 3, 8): 2048,
(0, 14, 3, 9): 4096,
(0, 14, 4, 8): 4096,
(0, 14, 4, 9): 8192,
(0, 14, 5, 8): 8192,
(0, 14, 5, 9): 16384,
(0, 14, 6, 8): 16384,
(0, 14, 6, 9): 32768,
(0, 14, 7, 8): 32768,
(0, 14, 7, 9): 65536,
(0, 14, 15, 2): 131072,
(0, 14, 15, 3): 262144,
(0, 14, 15, 4): 524288,
(0, 14, 15, 5): 1048576,
(0, 14, 15, 6): 2097152,
(0, 14, 15, 7): 4194304,
(0, 15): 12544,
(0, 15, 2, 8): 2048,
(0, 15, 2, 9): 4096,
(0, 15, 3, 8): 4096,
(0, 15, 3, 9): 8192,
(0, 15, 4, 8): 8192,
(0, 15, 4, 9): 16384,
(0, 15, 5, 8): 16384,
(0, 15, 5, 9): 32768,
(0, 15, 6, 8): 32768,
(0, 15, 6, 9): 65536,
(0, 15, 7, 8): 65536,
(0, 15, 7, 9): 131072,
(1, 2, 8): 16,
(1, 2, 8, 9): 64,
(1, 2, 9): 64,
(1, 2, 10): 256,
(1, 2, 11): 1024,
(1, 2, 12): 4096,
(1, 2, 13): 16384,
(1, 2, 14): 65536,
(1, 2, 15): 262144,
(1, 3, 8): 32,
(1, 3, 8, 9): 128,
(1, 3, 9): 128,
(1, 3, 10): 512,
(1, 3, 11): 2048,
(1, 3, 12): 8192,
(1, 3, 13): 32768,
(1, 3, 14): 131072,
(1, 3, 15): 524288,
(1, 4, 8): 64,
(1, 4, 8, 9): 256,
(1, 4, 9): 256,
(1, 4, 10): 1024,
(1, 4, 11): 4096,
(1, 4, 12): 16384,
(1, 4, 13): 65536,
(1, 4, 14): 262144,
(1, 4, 15): 1048576,
(1, 5, 8): 128,
(1, 5, 8, 9): 512,
(1, 5, 9): 512,
(1, 5, 10): 2048,
(1, 5, 11): 8192,
(1, 5, 12): 32768,
(1, 5, 13): 131072,
(1, 5, 14): 524288,
(1, 5, 15): 2097152,
(1, 6, 8): 256,
(1, 6, 8, 9): 1024,
(1, 6, 9): 1024,
(1, 6, 10): 4096,
(1, 6, 11): 16384,
(1, 6, 12): 65536,
(1, 6, 13): 262144,
(1, 6, 14): 1048576,
(1, 6, 15): 4194304,
(1, 7, 8): 512,
(1, 7, 8, 9): 2048,
(1, 7, 9): 2048,
(1, 7, 10): 8192,
(1, 7, 11): 32768,
(1, 7, 12): 131072,
(1, 7, 13): 524288,
(1, 7, 14): 2097152,
(1, 7, 15): 8388608,
(1, 8): -56,
(1, 9): -104,
(1, 10): -176,
(1, 10, 2, 8): 128,
(1, 10, 2, 9): 256,
(1, 10, 3, 8): 256,
(1, 10, 3, 9): 512,
(1, 10, 4, 8): 512,
(1, 10, 4, 9): 1024,
(1, 10, 5, 8): 1024,
(1, 10, 5, 9): 2048,
(1, 10, 6, 8): 2048,
(1, 10, 6, 9): 4096,
(1, 10, 7, 8): 4096,
(1, 10, 7, 9): 8192,
(1, 10, 11, 2): 1024,
(1, 10, 11, 3): 2048,
(1, 10, 11, 4): 4096,
(1, 10, 11, 5): 8192,
(1, 10, 11, 6): 16384,
(1, 10, 11, 7): 32768,
(1, 10, 12, 2): 2048,
(1, 10, 12, 3): 4096,
(1, 10, 12, 4): 8192,
(1, 10, 12, 5): 16384,
(1, 10, 12, 6): 32768,
(1, 10, 12, 7): 65536,
(1, 10, 13, 2): 4096,
(1, 10, 13, 3): 8192,
(1, 10, 13, 4): 16384,
(1, 10, 13, 5): 32768,
(1, 10, 13, 6): 65536,
(1, 10, 13, 7): 131072,
(1, 10, 14, 2): 8192,
(1, 10, 14, 3): 16384,
(1, 10, 14, 4): 32768,
(1, 10, 14, 5): 65536,
(1, 10, 14, 6): 131072,
(1, 10, 14, 7): 262144,
(1, 10, 15, 2): 16384,
(1, 10, 15, 3): 32768,
(1, 10, 15, 4): 65536,
(1, 10, 15, 5): 131072,
(1, 10, 15, 6): 262144,
(1, 10, 15, 7): 524288,
(1, 11): -224,
(1, 11, 2, 8): 256,
(1, 11, 2, 9): 512,
(1, 11, 3, 8): 512,
(1, 11, 3, 9): 1024,
(1, 11, 4, 8): 1024,
(1, 11, 4, 9): 2048,
(1, 11, 5, 8): 2048,
(1, 11, 5, 9): 4096,
(1, 11, 6, 8): 4096,
(1, 11, 6, 9): 8192,
(1, 11, 7, 8): 8192,
(1, 11, 7, 9): 16384,
(1, 11, 12, 2): 4096,
(1, 11, 12, 3): 8192,
(1, 11, 12, 4): 16384,
(1, 11, 12, 5): 32768,
(1, 11, 12, 6): 65536,
(1, 11, 12, 7): 131072,
(1, 11, 13, 2): 8192,
(1, 11, 13, 3): 16384,
(1, 11, 13, 4): 32768,
(1, 11, 13, 5): 65536,
(1, 11, 13, 6): 131072,
(1, 11, 13, 7): 262144,
(1, 11, 14, 2): 16384,
(1, 11, 14, 3): 32768,
(1, 11, 14, 4): 65536,
(1, 11, 14, 5): 131072,
(1, 11, 14, 6): 262144,
(1, 11, 14, 7): 524288,
(1, 11, 15, 2): 32768,
(1, 11, 15, 3): 65536,
(1, 11, 15, 4): 131072,
(1, 11, 15, 5): 262144,
(1, 11, 15, 6): 524288,
(1, 11, 15, 7): 1048576,
(1, 12): 64,
(1, 12, 2, 8): 512,
(1, 12, 2, 9): 1024,
(1, 12, 3, 8): 1024,
(1, 12, 3, 9): 2048,
(1, 12, 4, 8): 2048,
(1, 12, 4, 9): 4096,
(1, 12, 5, 8): 4096,
(1, 12, 5, 9): 8192,
(1, 12, 6, 8): 8192,
(1, 12, 6, 9): 16384,
(1, 12, 7, 8): 16384,
(1, 12, 7, 9): 32768,
(1, 12, 13, 2): 16384,
(1, 12, 13, 3): 32768,
(1, 12, 13, 4): 65536,
(1, 12, 13, 5): 131072,
(1, 12, 13, 6): 262144,
(1, 12, 13, 7): 524288,
(1, 12, 14, 2): 32768,
(1, 12, 14, 3): 65536,
(1, 12, 14, 4): 131072,
(1, 12, 14, 5): 262144,
(1, 12, 14, 6): 524288,
(1, 12, 14, 7): 1048576,
(1, 12, 15, 2): 65536,
(1, 12, 15, 3): 131072,
(1, 12, 15, 4): 262144,
(1, 12, 15, 5): 524288,
(1, 12, 15, 6): 1048576,
(1, 12, 15, 7): 2097152,
(1, 13): 2176,
(1, 13, 2, 8): 1024,
(1, 13, 2, 9): 2048,
(1, 13, 3, 8): 2048,
(1, 13, 3, 9): 4096,
(1, 13, 4, 8): 4096,
(1, 13, 4, 9): 8192,
(1, 13, 5, 8): 8192,
(1, 13, 5, 9): 16384,
(1, 13, 6, 8): 16384,
(1, 13, 6, 9): 32768,
(1, 13, 7, 8): 32768,
(1, 13, 7, 9): 65536,
(1, 13, 14, 2): 65536,
(1, 13, 14, 3): 131072,
(1, 13, 14, 4): 262144,
(1, 13, 14, 5): 524288,
(1, 13, 14, 6): 1048576,
(1, 13, 14, 7): 2097152,
(1, 13, 15, 2): 131072,
(1, 13, 15, 3): 262144,
(1, 13, 15, 4): 524288,
(1, 13, 15, 5): 1048576,
(1, 13, 15, 6): 2097152,
(1, 13, 15, 7): 4194304,
(1, 14): 12544,
(1, 14, 2, 8): 2048,
(1, 14, 2, 9): 4096,
(1, 14, 3, 8): 4096,
(1, 14, 3, 9): 8192,
(1, 14, 4, 8): 8192,
(1, 14, 4, 9): 16384,
(1, 14, 5, 8): 16384,
(1, 14, 5, 9): 32768,
(1, 14, 6, 8): 32768,
(1, 14, 6, 9): 65536,
(1, 14, 7, 8): 65536,
(1, 14, 7, 9): 131072,
(1, 14, 15, 2): 262144,
(1, 14, 15, 3): 524288,
(1, 14, 15, 4): 1048576,
(1, 14, 15, 5): 2097152,
(1, 14, 15, 6): 4194304,
(1, 14, 15, 7): 8388608,
(1, 15): 57856,
(1, 15, 2, 8): 4096,
(1, 15, 2, 9): 8192,
(1, 15, 3, 8): 8192,
(1, 15, 3, 9): 16384,
(1, 15, 4, 8): 16384,
(1, 15, 4, 9): 32768,
(1, 15, 5, 8): 32768,
(1, 15, 5, 9): 65536,
(1, 15, 6, 8): 65536,
(1, 15, 6, 9): 131072,
(1, 15, 7, 8): 131072,
(1, 15, 7, 9): 262144,
(2, 3, 8): 64,
(2, 3, 8, 9): 256,
(2, 3, 9): 256,
(2, 3, 10): 1024,
(2, 3, 11): 4096,
(2, 3, 12): 16384,
(2, 3, 13): 65536,
(2, 3, 14): 262144,
(2, 3, 15): 1048576,
(2, 4, 8): 128,
(2, 4, 8, 9): 512,
(2, 4, 9): 512,
(2, 4, 10): 2048,
(2, 4, 11): 8192,
(2, 4, 12): 32768,
(2, 4, 13): 131072,
(2, 4, 14): 524288,
(2, 4, 15): 2097152,
(2, 5, 8): 256,
(2, 5, 8, 9): 1024,
(2, 5, 9): 1024,
(2, 5, 10): 4096,
(2, 5, 11): 16384,
(2, 5, 12): 65536,
(2, 5, 13): 262144,
(2, 5, 14): 1048576,
(2, 5, 15): 4194304,
(2, 6, 8): 512,
(2, 6, 8, 9): 2048,
(2, 6, 9): 2048,
(2, 6, 10): 8192,
(2, 6, 11): 32768,
(2, 6, 12): 131072,
(2, 6, 13): 524288,
(2, 6, 14): 2097152,
(2, 6, 15): 8388608,
(2, 7, 8): 1024,
(2, 7, 8, 9): 4096,
(2, 7, 9): 4096,
(2, 7, 10): 16384,
(2, 7, 11): 65536,
(2, 7, 12): 262144,
(2, 7, 13): 1048576,
(2, 7, 14): 4194304,
(2, 7, 15): 16777216,
(2, 8): -104,
(2, 9): -176,
(3, 4, 8): 256,
(3, 4, 8, 9): 1024,
(3, 4, 9): 1024,
(3, 4, 10): 4096,
(3, 4, 11): 16384,
(3, 4, 12): 65536,
(3, 4, 13): 262144,
(3, 4, 14): 1048576,
(3, 4, 15): 4194304,
(3, 5, 8): 512,
(3, 5, 8, 9): 2048,
(3, 5, 9): 2048,
(3, 5, 10): 8192,
(3, 5, 11): 32768,
(3, 5, 12): 131072,
(3, 5, 13): 524288,
(3, 5, 14): 2097152,
(3, 5, 15): 8388608,
(3, 6, 8): 1024,
(3, 6, 8, 9): 4096,
(3, 6, 9): 4096,
(3, 6, 10): 16384,
(3, 6, 11): 65536,
(3, 6, 12): 262144,
(3, 6, 13): 1048576,
(3, 6, 14): 4194304,
(3, 6, 15): 16777216,
(3, 7, 8): 2048,
(3, 7, 8, 9): 8192,
(3, 7, 9): 8192,
(3, 7, 10): 32768,
(3, 7, 11): 131072,
(3, 7, 12): 524288,
(3, 7, 13): 2097152,
(3, 7, 14): 8388608,
(3, 7, 15): 33554432,
(3, 8): -176,
(3, 9): -224,
(4, 5, 8): 1024,
(4, 5, 8, 9): 4096,
(4, 5, 9): 4096,
(4, 5, 10): 16384,
(4, 5, 11): 65536,
(4, 5, 12): 262144,
(4, 5, 13): 1048576,
(4, 5, 14): 4194304,
(4, 5, 15): 16777216,
(4, 6, 8): 2048,
(4, 6, 8, 9): 8192,
(4, 6, 9): 8192,
(4, 6, 10): 32768,
(4, 6, 11): 131072,
(4, 6, 12): 524288,
(4, 6, 13): 2097152,
(4, 6, 14): 8388608,
(4, 6, 15): 33554432,
(4, 7, 8): 4096,
(4, 7, 8, 9): 16384,
(4, 7, 9): 16384,
(4, 7, 10): 65536,
(4, 7, 11): 262144,
(4, 7, 12): 1048576,
(4, 7, 13): 4194304,
(4, 7, 14): 16777216,
(4, 7, 15): 67108864,
(4, 8): -224,
(4, 9): 64,
(5, 6, 8): 4096,
(5, 6, 8, 9): 16384,
(5, 6, 9): 16384,
(5, 6, 10): 65536,
(5, 6, 11): 262144,
(5, 6, 12): 1048576,
(5, 6, 13): 4194304,
(5, 6, 14): 16777216,
(5, 6, 15): 67108864,
(5, 7, 8): 8192,
(5, 7, 8, 9): 32768,
(5, 7, 9): 32768,
(5, 7, 10): 131072,
(5, 7, 11): 524288,
(5, 7, 12): 2097152,
(5, 7, 13): 8388608,
(5, 7, 14): 33554432,
(5, 7, 15): 134217728,
(5, 8): 64,
(5, 9): 2176,
(6, 7, 8): 16384,
(6, 7, 8, 9): 65536,
(6, 7, 9): 65536,
(6, 7, 10): 262144,
(6, 7, 11): 1048576,
(6, 7, 12): 4194304,
(6, 7, 13): 16777216,
(6, 7, 14): 67108864,
(6, 7, 15): 268435456,
(6, 8): 2176,
(6, 9): 12544,
(7, 8): 12544,
(7, 9): 57856,
(8, 9, 0): 4,
(8, 9, 1): 16,
(8, 9, 2): 64,
(8, 9, 3): 256,
(8, 9, 4): 1024,
(8, 9, 5): 4096,
(8, 9, 6): 16384,
(8, 9, 7): 65536,
(10, 2): -224,
(10, 2, 3, 8): 512,
(10, 2, 3, 9): 1024,
(10, 2, 4, 8): 1024,
(10, 2, 4, 9): 2048,
(10, 2, 5, 8): 2048,
(10, 2, 5, 9): 4096,
(10, 2, 6, 8): 4096,
(10, 2, 6, 9): 8192,
(10, 2, 7, 8): 8192,
(10, 2, 7, 9): 16384,
(10, 3): 64,
(10, 3, 4, 8): 2048,
(10, 3, 4, 9): 4096,
(10, 3, 5, 8): 4096,
(10, 3, 5, 9): 8192,
(10, 3, 6, 8): 8192,
(10, 3, 6, 9): 16384,
(10, 3, 7, 8): 16384,
(10, 3, 7, 9): 32768,
(10, 4): 2176,
(10, 4, 5, 8): 8192,
(10, 4, 5, 9): 16384,
(10, 4, 6, 8): 16384,
(10, 4, 6, 9): 32768,
(10, 4, 7, 8): 32768,
(10, 4, 7, 9): 65536,
(10, 5): 12544,
(10, 5, 6, 8): 32768,
(10, 5, 6, 9): 65536,
(10, 5, 7, 8): 65536,
(10, 5, 7, 9): 131072,
(10, 6): 57856,
(10, 6, 7, 8): 131072,
(10, 6, 7, 9): 262144,
(10, 7): 246784,
(10, 8, 0): 8,
(10, 8, 1): 32,
(10, 8, 2): 128,
(10, 8, 3): 512,
(10, 8, 4): 2048,
(10, 8, 5): 8192,
(10, 8, 6): 32768,
(10, 8, 7): 131072,
(10, 9, 0): 16,
(10, 9, 1): 64,
(10, 9, 2): 256,
(10, 9, 3): 1024,
(10, 9, 4): 4096,
(10, 9, 5): 16384,
(10, 9, 6): 65536,
(10, 9, 7): 262144,
(10, 11, 0): 64,
(10, 11, 1): 256,
(10, 11, 2): 1024,
(10, 11, 2, 3): 4096,
(10, 11, 2, 4): 8192,
(10, 11, 2, 5): 16384,
(10, 11, 2, 6): 32768,
(10, 11, 2, 7): 65536,
(10, 11, 3): 4096,
(10, 11, 3, 4): 16384,
(10, 11, 3, 5): 32768,
(10, 11, 3, 6): 65536,
(10, 11, 3, 7): 131072,
(10, 11, 4): 16384,
(10, 11, 4, 5): 65536,
(10, 11, 4, 6): 131072,
(10, 11, 4, 7): 262144,
(10, 11, 5): 65536,
(10, 11, 5, 6): 262144,
(10, 11, 5, 7): 524288,
(10, 11, 6): 262144,
(10, 11, 6, 7): 1048576,
(10, 11, 7): 1048576,
(10, 12, 0): 128,
(10, 12, 1): 512,
(10, 12, 2): 2048,
(10, 12, 2, 3): 8192,
(10, 12, 2, 4): 16384,
(10, 12, 2, 5): 32768,
(10, 12, 2, 6): 65536,
(10, 12, 2, 7): 131072,
(10, 12, 3): 8192,
(10, 12, 3, 4): 32768,
(10, 12, 3, 5): 65536,
(10, 12, 3, 6): 131072,
(10, 12, 3, 7): 262144,
(10, 12, 4): 32768,
(10, 12, 4, 5): 131072,
(10, 12, 4, 6): 262144,
(10, 12, 4, 7): 524288,
(10, 12, 5): 131072,
(10, 12, 5, 6): 524288,
(10, 12, 5, 7): 1048576,
(10, 12, 6): 524288,
(10, 12, 6, 7): 2097152,
(10, 12, 7): 2097152,
(10, 13, 0): 256,
(10, 13, 1): 1024,
(10, 13, 2): 4096,
(10, 13, 2, 3): 16384,
(10, 13, 2, 4): 32768,
(10, 13, 2, 5): 65536,
(10, 13, 2, 6): 131072,
(10, 13, 2, 7): 262144,
(10, 13, 3): 16384,
(10, 13, 3, 4): 65536,
(10, 13, 3, 5): 131072,
(10, 13, 3, 6): 262144,
(10, 13, 3, 7): 524288,
(10, 13, 4): 65536,
(10, 13, 4, 5): 262144,
(10, 13, 4, 6): 524288,
(10, 13, 4, 7): 1048576,
(10, 13, 5): 262144,
(10, 13, 5, 6): 1048576,
(10, 13, 5, 7): 2097152,
(10, 13, 6): 1048576,
(10, 13, 6, 7): 4194304,
(10, 13, 7): 4194304,
(10, 14, 0): 512,
(10, 14, 1): 2048,
(10, 14, 2): 8192,
(10, 14, 2, 3): 32768,
(10, 14, 2, 4): 65536,
(10, 14, 2, 5): 131072,
(10, 14, 2, 6): 262144,
(10, 14, 2, 7): 524288,
(10, 14, 3): 32768,
(10, 14, 3, 4): 131072,
(10, 14, 3, 5): 262144,
(10, 14, 3, 6): 524288,
(10, 14, 3, 7): 1048576,
(10, 14, 4): 131072,
(10, 14, 4, 5): 524288,
(10, 14, 4, 6): 1048576,
(10, 14, 4, 7): 2097152,
(10, 14, 5): 524288,
(10, 14, 5, 6): 2097152,
(10, 14, 5, 7): 4194304,
(10, 14, 6): 2097152,
(10, 14, 6, 7): 8388608,
(10, 14, 7): 8388608,
(10, 15, 0): 1024,
(10, 15, 1): 4096,
(10, 15, 2): 16384,
(10, 15, 2, 3): 65536,
(10, 15, 2, 4): 131072,
(10, 15, 2, 5): 262144,
(10, 15, 2, 6): 524288,
(10, 15, 2, 7): 1048576,
(10, 15, 3): 65536,
(10, 15, 3, 4): 262144,
(10, 15, 3, 5): 524288,
(10, 15, 3, 6): 1048576,
(10, 15, 3, 7): 2097152,
(10, 15, 4): 262144,
(10, 15, 4, 5): 1048576,
(10, 15, 4, 6): 2097152,
(10, 15, 4, 7): 4194304,
(10, 15, 5): 1048576,
(10, 15, 5, 6): 4194304,
(10, 15, 5, 7): 8388608,
(10, 15, 6): 4194304,
(10, 15, 6, 7): 16777216,
(10, 15, 7): 16777216,
(11, 2): 64,
(11, 2, 3, 8): 1024,
(11, 2, 3, 9): 2048,
(11, 2, 4, 8): 2048,
(11, 2, 4, 9): 4096,
(11, 2, 5, 8): 4096,
(11, 2, 5, 9): 8192,
(11, 2, 6, 8): 8192,
(11, 2, 6, 9): 16384,
(11, 2, 7, 8): 16384,
(11, 2, 7, 9): 32768,
(11, 3): 2176,
(11, 3, 4, 8): 4096,
(11, 3, 4, 9): 8192,
(11, 3, 5, 8): 8192,
(11, 3, 5, 9): 16384,
(11, 3, 6, 8): 16384,
(11, 3, 6, 9): 32768,
(11, 3, 7, 8): 32768,
(11, 3, 7, 9): 65536,
(11, 4): 12544,
(11, 4, 5, 8): 16384,
(11, 4, 5, 9): 32768,
(11, 4, 6, 8): 32768,
(11, 4, 6, 9): 65536,
(11, 4, 7, 8): 65536,
(11, 4, 7, 9): 131072,
(11, 5): 57856,
(11, 5, 6, 8): 65536,
(11, 5, 6, 9): 131072,
(11, 5, 7, 8): 131072,
(11, 5, 7, 9): 262144,
(11, 6): 246784,
(11, 6, 7, 8): 262144,
(11, 6, 7, 9): 524288,
(11, 7): 1017856,
(11, 8, 0): 16,
(11, 8, 1): 64,
(11, 8, 2): 256,
(11, 8, 3): 1024,
(11, 8, 4): 4096,
(11, 8, 5): 16384,
(11, 8, 6): 65536,
(11, 8, 7): 262144,
(11, 9, 0): 32,
(11, 9, 1): 128,
(11, 9, 2): 512,
(11, 9, 3): 2048,
(11, 9, 4): 8192,
(11, 9, 5): 32768,
(11, 9, 6): 131072,
(11, 9, 7): 524288,
(11, 12, 0): 256,
(11, 12, 1): 1024,
(11, 12, 2): 4096,
(11, 12, 2, 3): 16384,
(11, 12, 2, 4): 32768,
(11, 12, 2, 5): 65536,
(11, 12, 2, 6): 131072,
(11, 12, 2, 7): 262144,
(11, 12, 3): 16384,
(11, 12, 3, 4): 65536,
(11, 12, 3, 5): 131072,
(11, 12, 3, 6): 262144,
(11, 12, 3, 7): 524288,
(11, 12, 4): 65536,
(11, 12, 4, 5): 262144,
(11, 12, 4, 6): 524288,
(11, 12, 4, 7): 1048576,
(11, 12, 5): 262144,
(11, 12, 5, 6): 1048576,
(11, 12, 5, 7): 2097152,
(11, 12, 6): 1048576,
(11, 12, 6, 7): 4194304,
(11, 12, 7): 4194304,
(11, 13, 0): 512,
(11, 13, 1): 2048,
(11, 13, 2): 8192,
(11, 13, 2, 3): 32768,
(11, 13, 2, 4): 65536,
(11, 13, 2, 5): 131072,
(11, 13, 2, 6): 262144,
(11, 13, 2, 7): 524288,
(11, 13, 3): 32768,
(11, 13, 3, 4): 131072,
(11, 13, 3, 5): 262144,
(11, 13, 3, 6): 524288,
(11, 13, 3, 7): 1048576,
(11, 13, 4): 131072,
(11, 13, 4, 5): 524288,
(11, 13, 4, 6): 1048576,
(11, 13, 4, 7): 2097152,
(11, 13, 5): 524288,
(11, 13, 5, 6): 2097152,
(11, 13, 5, 7): 4194304,
(11, 13, 6): 2097152,
(11, 13, 6, 7): 8388608,
(11, 13, 7): 8388608,
(11, 14, 0): 1024,
(11, 14, 1): 4096,
(11, 14, 2): 16384,
(11, 14, 2, 3): 65536,
(11, 14, 2, 4): 131072,
(11, 14, 2, 5): 262144,
(11, 14, 2, 6): 524288,
(11, 14, 2, 7): 1048576,
(11, 14, 3): 65536,
(11, 14, 3, 4): 262144,
(11, 14, 3, 5): 524288,
(11, 14, 3, 6): 1048576,
(11, 14, 3, 7): 2097152,
(11, 14, 4): 262144,
(11, 14, 4, 5): 1048576,
(11, 14, 4, 6): 2097152,
(11, 14, 4, 7): 4194304,
(11, 14, 5): 1048576,
(11, 14, 5, 6): 4194304,
(11, 14, 5, 7): 8388608,
(11, 14, 6): 4194304,
(11, 14, 6, 7): 16777216,
(11, 14, 7): 16777216,
(11, 15, 0): 2048,
(11, 15, 1): 8192,
(11, 15, 2): 32768,
(11, 15, 2, 3): 131072,
(11, 15, 2, 4): 262144,
(11, 15, 2, 5): 524288,
(11, 15, 2, 6): 1048576,
(11, 15, 2, 7): 2097152,
(11, 15, 3): 131072,
(11, 15, 3, 4): 524288,
(11, 15, 3, 5): 1048576,
(11, 15, 3, 6): 2097152,
(11, 15, 3, 7): 4194304,
(11, 15, 4): 524288,
(11, 15, 4, 5): 2097152,
(11, 15, 4, 6): 4194304,
(11, 15, 4, 7): 8388608,
(11, 15, 5): 2097152,
(11, 15, 5, 6): 8388608,
(11, 15, 5, 7): 16777216,
(11, 15, 6): 8388608,
(11, 15, 6, 7): 33554432,
(11, 15, 7): 33554432,
(12, 2): 2176,
(12, 2, 3, 8): 2048,
(12, 2, 3, 9): 4096,
(12, 2, 4, 8): 4096,
(12, 2, 4, 9): 8192,
(12, 2, 5, 8): 8192,
(12, 2, 5, 9): 16384,
(12, 2, 6, 8): 16384,
(12, 2, 6, 9): 32768,
(12, 2, 7, 8): 32768,
(12, 2, 7, 9): 65536,
(12, 3): 12544,
(12, 3, 4, 8): 8192,
(12, 3, 4, 9): 16384,
(12, 3, 5, 8): 16384,
(12, 3, 5, 9): 32768,
(12, 3, 6, 8): 32768,
(12, 3, 6, 9): 65536,
(12, 3, 7, 8): 65536,
(12, 3, 7, 9): 131072,
(12, 4): 57856,
(12, 4, 5, 8): 32768,
(12, 4, 5, 9): 65536,
(12, 4, 6, 8): 65536,
(12, 4, 6, 9): 131072,
(12, 4, 7, 8): 131072,
(12, 4, 7, 9): 262144,
(12, 5): 246784,
(12, 5, 6, 8): 131072,
(12, 5, 6, 9): 262144,
(12, 5, 7, 8): 262144,
(12, 5, 7, 9): 524288,
(12, 6): 1017856,
(12, 6, 7, 8): 524288,
(12, 6, 7, 9): 1048576,
(12, 7): 4132864,
(12, 8, 0): 32,
(12, 8, 1): 128,
(12, 8, 2): 512,
(12, 8, 3): 2048,
(12, 8, 4): 8192,
(12, 8, 5): 32768,
(12, 8, 6): 131072,
(12, 8, 7): 524288,
(12, 9, 0): 64,
(12, 9, 1): 256,
(12, 9, 2): 1024,
(12, 9, 3): 4096,
(12, 9, 4): 16384,
(12, 9, 5): 65536,
(12, 9, 6): 262144,
(12, 9, 7): 1048576,
(12, 13, 0): 1024,
(12, 13, 1): 4096,
(12, 13, 2): 16384,
(12, 13, 2, 3): 65536,
(12, 13, 2, 4): 131072,
(12, 13, 2, 5): 262144,
(12, 13, 2, 6): 524288,
(12, 13, 2, 7): 1048576,
(12, 13, 3): 65536,
(12, 13, 3, 4): 262144,
(12, 13, 3, 5): 524288,
(12, 13, 3, 6): 1048576,
(12, 13, 3, 7): 2097152,
(12, 13, 4): 262144,
(12, 13, 4, 5): 1048576,
(12, 13, 4, 6): 2097152,
(12, 13, 4, 7): 4194304,
(12, 13, 5): 1048576,
(12, 13, 5, 6): 4194304,
(12, 13, 5, 7): 8388608,
(12, 13, 6): 4194304,
(12, 13, 6, 7): 16777216,
(12, 13, 7): 16777216,
(12, 14, 0): 2048,
(12, 14, 1): 8192,
(12, 14, 2): 32768,
(12, 14, 2, 3): 131072,
(12, 14, 2, 4): 262144,
(12, 14, 2, 5): 524288,
(12, 14, 2, 6): 1048576,
(12, 14, 2, 7): 2097152,
(12, 14, 3): 131072,
(12, 14, 3, 4): 524288,
(12, 14, 3, 5): 1048576,
(12, 14, 3, 6): 2097152,
(12, 14, 3, 7): 4194304,
(12, 14, 4): 524288,
(12, 14, 4, 5): 2097152,
(12, 14, 4, 6): 4194304,
(12, 14, 4, 7): 8388608,
(12, 14, 5): 2097152,
(12, 14, 5, 6): 8388608,
(12, 14, 5, 7): 16777216,
(12, 14, 6): 8388608,
(12, 14, 6, 7): 33554432,
(12, 14, 7): 33554432,
(12, 15, 0): 4096,
(12, 15, 1): 16384,
(12, 15, 2): 65536,
(12, 15, 2, 3): 262144,
(12, 15, 2, 4): 524288,
(12, 15, 2, 5): 1048576,
(12, 15, 2, 6): 2097152,
(12, 15, 2, 7): 4194304,
(12, 15, 3): 262144,
(12, 15, 3, 4): 1048576,
(12, 15, 3, 5): 2097152,
(12, 15, 3, 6): 4194304,
(12, 15, 3, 7): 8388608,
(12, 15, 4): 1048576,
(12, 15, 4, 5): 4194304,
(12, 15, 4, 6): 8388608,
(12, 15, 4, 7): 16777216,
(12, 15, 5): 4194304,
(12, 15, 5, 6): 16777216,
(12, 15, 5, 7): 33554432,
(12, 15, 6): 16777216,
(12, 15, 6, 7): 67108864,
(12, 15, 7): 67108864,
(13, 2): 12544,
(13, 2, 3, 8): 4096,
(13, 2, 3, 9): 8192,
(13, 2, 4, 8): 8192,
(13, 2, 4, 9): 16384,
(13, 2, 5, 8): 16384,
(13, 2, 5, 9): 32768,
(13, 2, 6, 8): 32768,
(13, 2, 6, 9): 65536,
(13, 2, 7, 8): 65536,
(13, 2, 7, 9): 131072,
(13, 3): 57856,
(13, 3, 4, 8): 16384,
(13, 3, 4, 9): 32768,
(13, 3, 5, 8): 32768,
(13, 3, 5, 9): 65536,
(13, 3, 6, 8): 65536,
(13, 3, 6, 9): 131072,
(13, 3, 7, 8): 131072,
(13, 3, 7, 9): 262144,
(13, 4): 246784,
(13, 4, 5, 8): 65536,
(13, 4, 5, 9): 131072,
(13, 4, 6, 8): 131072,
(13, 4, 6, 9): 262144,
(13, 4, 7, 8): 262144,
(13, 4, 7, 9): 524288,
(13, 5): 1017856,
(13, 5, 6, 8): 262144,
(13, 5, 6, 9): 524288,
(13, 5, 7, 8): 524288,
(13, 5, 7, 9): 1048576,
(13, 6): 4132864,
(13, 6, 7, 8): 1048576,
(13, 6, 7, 9): 2097152,
(13, 7): 16654336,
(13, 8, 0): 64,
(13, 8, 1): 256,
(13, 8, 2): 1024,
(13, 8, 3): 4096,
(13, 8, 4): 16384,
(13, 8, 5): 65536,
(13, 8, 6): 262144,
(13, 8, 7): 1048576,
(13, 9, 0): 128,
(13, 9, 1): 512,
(13, 9, 2): 2048,
(13, 9, 3): 8192,
(13, 9, 4): 32768,
(13, 9, 5): 131072,
(13, 9, 6): 524288,
(13, 9, 7): 2097152,
(13, 14, 0): 4096,
(13, 14, 1): 16384,
(13, 14, 2): 65536,
(13, 14, 2, 3): 262144,
(13, 14, 2, 4): 524288,
(13, 14, 2, 5): 1048576,
(13, 14, 2, 6): 2097152,
(13, 14, 2, 7): 4194304,
(13, 14, 3): 262144,
(13, 14, 3, 4): 1048576,
(13, 14, 3, 5): 2097152,
(13, 14, 3, 6): 4194304,
(13, 14, 3, 7): 8388608,
(13, 14, 4): 1048576,
(13, 14, 4, 5): 4194304,
(13, 14, 4, 6): 8388608,
(13, 14, 4, 7): 16777216,
(13, 14, 5): 4194304,
(13, 14, 5, 6): 16777216,
(13, 14, 5, 7): 33554432,
(13, 14, 6): 16777216,
(13, 14, 6, 7): 67108864,
(13, 14, 7): 67108864,
(13, 15, 0): 8192,
(13, 15, 1): 32768,
(13, 15, 2): 131072,
(13, 15, 2, 3): 524288,
(13, 15, 2, 4): 1048576,
(13, 15, 2, 5): 2097152,
(13, 15, 2, 6): 4194304,
(13, 15, 2, 7): 8388608,
(13, 15, 3): 524288,
(13, 15, 3, 4): 2097152,
(13, 15, 3, 5): 4194304,
(13, 15, 3, 6): 8388608,
(13, 15, 3, 7): 16777216,
(13, 15, 4): 2097152,
(13, 15, 4, 5): 8388608,
(13, 15, 4, 6): 16777216,
(13, 15, 4, 7): 33554432,
(13, 15, 5): 8388608,
(13, 15, 5, 6): 33554432,
(13, 15, 5, 7): 67108864,
(13, 15, 6): 33554432,
(13, 15, 6, 7): 134217728,
(13, 15, 7): 134217728,
(14, 2): 57856,
(14, 2, 3, 8): 8192,
(14, 2, 3, 9): 16384,
(14, 2, 4, 8): 16384,
(14, 2, 4, 9): 32768,
(14, 2, 5, 8): 32768,
(14, 2, 5, 9): 65536,
(14, 2, 6, 8): 65536,
(14, 2, 6, 9): 131072,
(14, 2, 7, 8): 131072,
(14, 2, 7, 9): 262144,
(14, 3): 246784,
(14, 3, 4, 8): 32768,
(14, 3, 4, 9): 65536,
(14, 3, 5, 8): 65536,
(14, 3, 5, 9): 131072,
(14, 3, 6, 8): 131072,
(14, 3, 6, 9): 262144,
(14, 3, 7, 8): 262144,
(14, 3, 7, 9): 524288,
(14, 4): 1017856,
(14, 4, 5, 8): 131072,
(14, 4, 5, 9): 262144,
(14, 4, 6, 8): 262144,
(14, 4, 6, 9): 524288,
(14, 4, 7, 8): 524288,
(14, 4, 7, 9): 1048576,
(14, 5): 4132864,
(14, 5, 6, 8): 524288,
(14, 5, 6, 9): 1048576,
(14, 5, 7, 8): 1048576,
(14, 5, 7, 9): 2097152,
(14, 6): 16654336,
(14, 6, 7, 8): 2097152,
(14, 6, 7, 9): 4194304,
(14, 7): 66863104,
(14, 8, 0): 128,
(14, 8, 1): 512,
(14, 8, 2): 2048,
(14, 8, 3): 8192,
(14, 8, 4): 32768,
(14, 8, 5): 131072,
(14, 8, 6): 524288,
(14, 8, 7): 2097152,
(14, 9, 0): 256,
(14, 9, 1): 1024,
(14, 9, 2): 4096,
(14, 9, 3): 16384,
(14, 9, 4): 65536,
(14, 9, 5): 262144,
(14, 9, 6): 1048576,
(14, 9, 7): 4194304,
(14, 15, 0): 16384,
(14, 15, 1): 65536,
(14, 15, 2): 262144,
(14, 15, 2, 3): 1048576,
(14, 15, 2, 4): 2097152,
(14, 15, 2, 5): 4194304,
(14, 15, 2, 6): 8388608,
(14, 15, 2, 7): 16777216,
(14, 15, 3): 1048576,
(14, 15, 3, 4): 4194304,
(14, 15, 3, 5): 8388608,
(14, 15, 3, 6): 16777216,
(14, 15, 3, 7): 33554432,
(14, 15, 4): 4194304,
(14, 15, 4, 5): 16777216,
(14, 15, 4, 6): 33554432,
(14, 15, 4, 7): 67108864,
(14, 15, 5): 16777216,
(14, 15, 5, 6): 67108864,
(14, 15, 5, 7): 134217728,
(14, 15, 6): 67108864,
(14, 15, 6, 7): 268435456,
(14, 15, 7): 268435456,
(15, 2): 246784,
(15, 2, 3, 8): 16384,
(15, 2, 3, 9): 32768,
(15, 2, 4, 8): 32768,
(15, 2, 4, 9): 65536,
(15, 2, 5, 8): 65536,
(15, 2, 5, 9): 131072,
(15, 2, 6, 8): 131072,
(15, 2, 6, 9): 262144,
(15, 2, 7, 8): 262144,
(15, 2, 7, 9): 524288,
(15, 3): 1017856,
(15, 3, 4, 8): 65536,
(15, 3, 4, 9): 131072,
(15, 3, 5, 8): 131072,
(15, 3, 5, 9): 262144,
(15, 3, 6, 8): 262144,
(15, 3, 6, 9): 524288,
(15, 3, 7, 8): 524288,
(15, 3, 7, 9): 1048576,
(15, 4): 4132864,
(15, 4, 5, 8): 262144,
(15, 4, 5, 9): 524288,
(15, 4, 6, 8): 524288,
(15, 4, 6, 9): 1048576,
(15, 4, 7, 8): 1048576,
(15, 4, 7, 9): 2097152,
(15, 5): 16654336,
(15, 5, 6, 8): 1048576,
(15, 5, 6, 9): 2097152,
(15, 5, 7, 8): 2097152,
(15, 5, 7, 9): 4194304,
(15, 6): 66863104,
(15, 6, 7, 8): 4194304,
(15, 6, 7, 9): 8388608,
(15, 7): 267943936,
(15, 8, 0): 256,
(15, 8, 1): 1024,
(15, 8, 2): 4096,
(15, 8, 3): 16384,
(15, 8, 4): 65536,
(15, 8, 5): 262144,
(15, 8, 6): 1048576,
(15, 8, 7): 4194304,
(15, 9, 0): 512,
(15, 9, 1): 2048,
(15, 9, 2): 8192,
(15, 9, 3): 32768,
(15, 9, 4): 131072,
(15, 9, 5): 524288,
(15, 9, 6): 2097152,
(15, 9, 7): 8388608
}
```

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