Buggy python submission (verdict: Time Limit Exceeded):
# coding: utf-8
import sys
#from operator import itemgetter
sysread = sys.stdin.buffer.readline
read = sys.stdin.buffer.read
#from heapq import heappop, heappush
#from collections import defaultdict
sys.setrecursionlimit(10**7)
#import math
#from itertools import product, accumulate, combinations, product
#import bisect# lower_bound etc
#import numpy as np
#from copy import deepcopy
#from collections import deque
import math

def run():
    N = int(input())
    for n in range(1, N+1):
        ret = 0
        val = int(math.sqrt(2 * n))
        for X in range(-val, val+1):
            for Y in range(-val, val+1):
                tmp1 = 2 * n - X ** 2 - Y ** 2
                if tmp1 >= 0 and math.sqrt(tmp1) == int(math.sqrt(tmp1)):
                    Z_ = int(math.sqrt(tmp1))
                    if Z_ != 0:
                        for k in [-1, 1]:
                            Z = Z_ * k
                            if not (X + Y + Z) % 2:
                                xyz = (X + Y + Z) // 2
                                x = xyz - Y
                                y = xyz - Z
                                z = xyz - X
                                if x >= 1 and y >= 1 and z >= 1:
                                    ret += 1
                                    #print(x, y,z)


        print(ret)



if __name__ == "__main__":
    run()


Fixed submission:
# coding: utf-8
import sys
#from operator import itemgetter
sysread = sys.stdin.buffer.readline
read = sys.stdin.buffer.read
#from heapq import heappop, heappush
#from collections import defaultdict
sys.setrecursionlimit(10**7)
#import math
#from itertools import product, accumulate, combinations, product
#import bisect# lower_bound etc
#import numpy as np
#from copy import deepcopy
#from collections import deque
import math

def run():
    N = int(input())
    for n in range(1, N+1):
        ret = 0
        val = int(math.sqrt(2 * n))
        for X in range(2, val+1):
            for Y in range(2, val+1):
                tmp1 = 2 * n - X ** 2 - Y ** 2
                if tmp1 >= 0 and math.sqrt(tmp1) == int(math.sqrt(tmp1)):
                    Z_ = int(math.sqrt(tmp1))
                    if Z_ != 0:
                        for k in [-1, 1]:
                            Z = Z_ * k
                            if not (X + Y + Z) % 2:
                                xyz = (X + Y + Z) // 2
                                x = xyz - Y
                                y = xyz - Z
                                z = xyz - X
                                if x >= 1 and y >= 1 and z >= 1:
                                    ret += 1
                                    #print(x, y,z)


        print(ret)



if __name__ == "__main__":
    run()

Buggy python submission (verdict: Time Limit Exceeded):
# coding: utf-8
import sys
#from operator import itemgetter
sysread = sys.stdin.buffer.readline
read = sys.stdin.buffer.read
#from heapq import heappop, heappush
#from collections import defaultdict
sys.setrecursionlimit(10**7)
#import math
#from itertools import product, accumulate, combinations, product
#import bisect# lower_bound etc
#import numpy as np
#from copy import deepcopy
#from collections import deque
import math

def run():
    N = int(input())
    for n in range(1, N+1):
        ret = 0
        val = int(math.log2(2 * n))
        for X in range(-val, val+1):
            for Y in range(-val, val+1):
                tmp1 = 2 * n - X ** 2 - Y ** 2
                if tmp1 >= 0 and math.sqrt(tmp1) == int(math.sqrt(tmp1)):
                    Z_ = int(math.sqrt(tmp1))
                    if Z_ != 0:
                        for k in [-1, 1]:
                            Z = Z_ * k
                            if not (X + Y + Z) % 2:
                                xyz = (X + Y + Z) // 2
                                x = xyz - Y
                                y = xyz - Z
                                z = xyz - X
                                if x >= 1 and y >= 1 and z >= 1:
                                    ret += 1
                                    #print(x, y,z)
                    else:
                        if not (X + Y + Z_) % 2:
                            xyz = (X + Y + Z_) // 2
                            x = xyz - Y
                            y = xyz - Z_
                            z = xyz - X
                            if x >= 1 and y >= 1 and z >= 1:
                                ret += 1

        print(ret)



if __name__ == "__main__":
    run()


Fixed submission:
# coding: utf-8
import sys
#from operator import itemgetter
sysread = sys.stdin.buffer.readline
read = sys.stdin.buffer.read
#from heapq import heappop, heappush
#from collections import defaultdict
sys.setrecursionlimit(10**7)
#import math
#from itertools import product, accumulate, combinations, product
#import bisect# lower_bound etc
#import numpy as np
#from copy import deepcopy
#from collections import deque
import math

def run():
    N = int(input())
    for n in range(1, N+1):
        ret = 0
        val = int(math.sqrt(2 * n))
        for X in range(2, val+1):
            for Y in range(2, val+1):
                tmp1 = 2 * n - X ** 2 - Y ** 2
                if tmp1 >= 0 and math.sqrt(tmp1) == int(math.sqrt(tmp1)):
                    Z_ = int(math.sqrt(tmp1))
                    if Z_ != 0:
                        for k in [-1, 1]:
                            Z = Z_ * k
                            if not (X + Y + Z) % 2:
                                xyz = (X + Y + Z) // 2
                                x = xyz - Y
                                y = xyz - Z
                                z = xyz - X
                                if x >= 1 and y >= 1 and z >= 1:
                                    ret += 1
                                    #print(x, y,z)


        print(ret)



if __name__ == "__main__":
    run()