# python 3+

# first task:

from collections import Counter

# below is algorithm, that show idea

def is_prime(n):

   for num in range(2, n):

       if n % num == 0: return False

   return True

def next_prime_from(n):

   if is_prime(n): n += 1

   while True:

       if is_prime(n): return n

       n += 1

def get_multipliers(n):

   multipliers = []

   remainder = None

   divider = 2

   while remainder != 1:

       if n % divider == 0:

           remainder = n / divider

           multipliers.append(divider)

           n /= divider

           continue

       divider = next_prime_from(divider)

   return multipliers

def canonically_decompose(n):

   multipliers = get_multipliers(n)

   decomposed = Counter(multipliers)

   return decomposed

def get_divigions_count(n):

   prediction = 1

   degrees = []

   nums = canonically_decompose(n)

   nna = set(get_multipliers(n))

   for element in nna: degrees.append(nums[element])

   for count in degrees: prediction *= count+1

   return prediction

# above is algorithm, that show idea

# you can play with it

# simple change of numbers

def get_dividers(n):

   dividers = []

   for num in range(n, 0, -1):

       if (n % num == 0):

           dividers.append(num)

   return dividers

x = list(range(81234, 134689 + 1)) # sometimes it so slow. so I left you commented list below. You can try those list. The result is the same.

# x = [16, 81, 54, 12, 625, 45, 22, 96] # very fast

# x = [i for i in range(0, 1000)] # fast

numbers = []

for num in x:

   if len(numbers) > 2: break

   if str(num ** (0.5)).split(".")[1] != "0": continue

   if len(get_dividers(num)) == 5: numbers.append(num)

for element in sorted(numbers): print(f"{element}: {sorted(get_dividers(element))}")

# the output will be 83521: [1, 17, 289, 4913, 83521]\n130321: [1, 19, 361, 6859, 130321]. thats because in those numbers interval no satisfying numbers.

# you can turn "x" above. and you'll see, that program works correctly

input("\nTap on Enter to continue...")

# second task:

def get_dividers(n):

   dividers = []

   for num in range(2, n-1):

       if n % num == 0: dividers.append(num)

   return dividers

x = 250200

predictions = []

while len(predictions) + 1 < 6: # you need wait a little bit. it continue on 250704

   x += 1

   dividers = get_dividers(x)

   try:

       if (max(dividers) + min(dividers)) % 123 == 17: predictions.append(x)

   except ValueError: continue

for element in predictions:

   dividers = get_dividers(element)

   try: print(": ".join(list(map(str, [element, min(dividers) + max(dividers)]))))

   except: continue

input("\nTap on Enter to continue...")

# if something doesn't work, you can write it in the comments, and i will answer you, and other ;)