We are interested in collecting the triples of positive integers ```(a, b, c)``` that fulfill the following equation:
```python
a² + b² = c³
```
The first triple with the lowest values that satisfies the equation we have above is (2, 2 ,2).
In effect:
```python
2² + 2² = 2³
4  + 4  = 8
```
The first pair of triples that "shares" the same value of ```c``` is: ```(2, 11, 5)``` and ```(5, 10, 5)```. 

Both triples share the same value of ```c```  is ```c = 5```. 
```python
Triple (2, 11, 5)                  Triple(5, 10, 5)
2² + 11² =  5³                      5² +  10²  =  5³
4  + 121 = 125                     25  + 100   = 125
```
So, we say that the value ```c``` has two solutions because there are two triples sharing the same value of ```c```.

There are some values of ```c``` with no solutions.

The first value of ```c``` that have a surprising number of solutions is ```65``` with ```8``` different triples.

In order to avoid duplications you will consider that ```a <= b``` always.

Make the function ```find_abc_sumsqcube()```, that may give us the values of c for an specific number of solutions.

For that purpose the above required function will receive two arguments, ```c_max``` and ```num_sol```. It is understandable that ```c_max``` will give to our function the upper limit of ```c``` and ```num_sol```, the specific number of solutions.

The function will output a sorted list with the values of ```c``` that have a number of solutions equals to ```num_sol```

Let's see some cases: 
```python
find_abc_sumsqcube(5, 1) == [2] # below or equal to c_max = 5 we have triple the (2, 2, 2) (see above)

find_abc_sumsqcube(5, 2) == [5] # now we want the values of ```c ≤ c_max``` with two solutions (see above again)

find_abc_sumsqcube(10, 2) == [5, 10]

find_abc_sumsqcube(20, 8) == [] # There are no values of c equal and bellow 20 having 8 solutions.
```

Our tests will have the following ranges for our two arguments:
```python
5 ≤ c_max ≤ 1000
1 ≤ num_sol ≤ 10
```
Happy coding!!

```python

USER_FILENAME = "<user_code>"
import sys, json, builtins
import math, heapq, bisect, itertools, functools, collections, statistics, fractions, decimal, operator, array, random, re, string, os, pathlib
try:
    import resource as _res
except Exception:
    _res = None
from collections import Counter, defaultdict, deque, OrderedDict
from functools import lru_cache, reduce
from itertools import accumulate, permutations, combinations, product
from math import gcd, ceil, floor, sqrt
from decimal import Decimal
try:
    from typing import *
except Exception:
    pass
try:
    sys.setrecursionlimit(1_000_000)
except Exception:
    pass
try:
    import site as _site
    try:
        _sys_sites = _site.getsitepackages() if hasattr(_site, "getsitepackages") else []
        for _p in _sys_sites:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
    try:
        _usp = _site.getusersitepackages()
        if isinstance(_usp, str):
            _usp = [_usp]
        for _p in _usp:
            if _p and _p not in sys.path:
                sys.path.append(_p)
    except Exception:
        pass
except Exception:
    pass

def _disabled_input(*args, **kwargs):
    raise RuntimeError("input() is disabled in callable mode")
builtins.input = _disabled_input

ns = {"__name__": "__judge__"}  # prevent if __name__ == "__main__" blocks
for _k, _v in list(globals().items()):
    if not _k.startswith("_"):
        ns[_k] = _v
_CODE = 'import math\nfrom functools import lru_cache\n\n@lru_cache(maxsize = 1024)\ndef solutions(c):\n    cube = c ** 3\n    return sum(1 for a in range(1,int((cube//2)**.5)+1) if ((cube-a**2)**.5).is_integer())\n\n\ndef find_abc_sumsqcube(c_max, num_sol):\n    result = []\n    for c in range(2,c_max+1):\n        if solutions(c) == num_sol:\n            result.append(c)\n    return result\n'
_co = compile(_CODE, USER_FILENAME, "exec")
exec(_co, ns, ns)

_target = None
_fn_name = 'find_abc_sumsqcube'
if _fn_name:
    if "Solution" in ns:
        try:
            _obj = ns["Solution"]()
            if hasattr(_obj, _fn_name):
                _target = getattr(_obj, _fn_name)
        except Exception:
            _target = None
    if _target is None and _fn_name in ns and callable(ns[_fn_name]):
        _target = ns[_fn_name]
# Fallback: if there's exactly one user-defined function, use it
if _target is None:
    _cands = [v for k, v in ns.items() if callable(v) and getattr(v, "__module__", "") in ("__judge__", "__main__")]
    _user_funcs = [c for c in _cands if getattr(c, "__name__", "").startswith("_") is False]
    if len(_user_funcs) == 1:
        _target = _user_funcs[0]
if _target is None:
    raise RuntimeError(f"Could not resolve callable '{_fn_name}'")

_raw = sys.stdin.read().strip()
if _raw:
    _parsed = json.loads(_raw)
    if isinstance(_parsed, dict):
        _kwargs = _parsed.get("kwargs", {})
        _args = _parsed.get("args", [])
    else:
        _args, _kwargs = _parsed, {}
else:
    _args, _kwargs = [], {}
if not isinstance(_args, (list, tuple)):
    _args = [_args]
_res = _target(*_args, **_kwargs)
try:
    sys.stdout.write(repr(_res) + "\n")
except Exception:
    pass

```

Measured memory_bytes: 6492