Problem p03253 (python)

Fixed submission:
import math
def fact(a,b):
    ans = 1
    while a != b:
        ans *= a
        a -= 1
    return ans
def nCr(n,r):
    return (fact(n,r)) // (math.factorial(n-r))
def nHr(n,r):
    return nCr(n+r-1, r-1)
def prime(n): # nまでの素数を列挙
    import math
    num_list = [i + 1 for i in range(2,n,2)]
    list_prime = [2]
    limit = math.sqrt(n)
    if n == 2:
        return list_prime
    else:
        while True:
            p = num_list[0]
            if p >= limit:
                return list_prime + num_list
            list_prime.append(p)
            num_list = [num for num in num_list if num % p != 0]

def primeFactorization(n):
    import math
    list_prime = prime(int(math.sqrt(n)))
    i = 0
    dict_pF = {}
    dict_primeFactorization = {}
    for pri in list_prime:
        dict_pF[pri] = 0
    while True:
        if n == 1:
            for key, value in dict_pF.items():
                if value != 0:
                    dict_primeFactorization[key] = value
            return dict_primeFactorization
        elif i >= len(list_prime):
            dict_pF[n] = 1
            for key, value in dict_pF.items():
                if value != 0:
                    dict_primeFactorization[key] = value
            return dict_primeFactorization
        p = list_prime[i]
        if n % p == 0:
            n //= p
            dict_pF[p] += 1
            continue
        else:
            i += 1

ans = 1
n,m = map(int, input().split())
if m == 1:
    ans = 1
else:
    for factor,degree in primeFactorization(m).items():
        ans *= (nHr(degree,n)) % (10**9+7)
        
print(ans % (10**9+7))

Buggy submission (verdict: Runtime Error):
import math
def nCr(n,r):
    return math.factorial(n) / math.factorial(r) / math.factorial(n-r)
def nHr(n,r):
    return nCr(n+r-1, r-1)
def prime(n): # nまでの素数を列挙
    import math
    num_list = [i + 1 for i in range(2,n,2)]
    list_prime = [2]
    limit = math.sqrt(n)
    if n == 2:
        return list_prime
    else:
        while True:
            p = num_list[0]
            if p >= limit:
                return list_prime + num_list
            list_prime.append(p)
            num_list = [num for num in num_list if num % p != 0]

def primeFactorization(n):
    import math
    list_prime = prime(int(math.sqrt(n)))
    i = 0
    dict_pF = {}
    dict_primeFactorization = {}
    for pri in list_prime:
        dict_pF[pri] = 0
    while True:
        if n == 1:
            for key, value in dict_pF.items():
                if value != 0:
                    dict_primeFactorization[key] = value
            return dict_primeFactorization
        elif i >= len(list_prime):
            dict_pF[n] = 1
            for key, value in dict_pF.items():
                if value != 0:
                    dict_primeFactorization[key] = value
            return dict_primeFactorization
        p = list_prime[i]
        if n % p == 0:
            n //= p
            dict_pF[p] += 1
            continue
        else:
            i += 1

ans = 1
n,m = map(int, input().split())
for factor,degree in primeFactorization(m).items():
    ans *= nHr(degree,n)

print(int(ans) % (10**9+7))

Buggy submission (verdict: Runtime Error):
def factorial(n):
	fact = 1
	while n > 0:
		fact *= n
		n -= 1
	return fact
	
def combination(n, r):
    return factorial(n) // (factorial(n - r) * factorial(r))

n, m = map(int, input().split())
n_def = n
natural = list(range(2,320))
index = 0
while index < len(natural):
	for number in natural:
		if number % natural[index] == 0 and number / natural[index] != 1:
			natural.remove(number)
	index += 1
prime = natural

prime_division = {}
index = 0
number_prime = 0
while prime[index] <= m:
	if m % prime[index] == 0:
		m /= prime[index]
		number_prime += 1
		continue		
	else:
		prime_division[prime[index]] = number_prime
		number_prime = 0
		index += 1
else:
	prime_division[prime[index]] = number_prime

list_division = list(prime_division.values())
ans = 1
for i in list_division:
	ans *= combination(i+n-1,n-1)
print(int(ans%(10**9 + 7)))

Buggy submission (verdict: Runtime Error):
import math
def combination(n, r):
    return math.factorial(n) // (math.factorial(n - r) * math.factorial(r))
n, m = map(int, input().split())
n_def = n
natural = list(range(2,400))
index = 0
while index < len(natural):
	for number in natural:
		if number % natural[index] == 0 and number / natural[index] != 1:
			natural.remove(number)
	index += 1
prime = natural

prime_division = {}
index = 0
number_prime = 0
while prime[index] <= m:
	if m % prime[index] == 0:
		m /= prime[index]
		number_prime += 1
		continue		
	else:
		prime_division[prime[index]] = number_prime
		number_prime = 0
		index += 1
else:
	prime_division[prime[index]] = number_prime

list_division = list(prime_division.values())
ans = 1
for i in list_division:
#print(combination(i+n-1,n-1))
	ans *= combination(i+n-1,n-1)
print(int(ans%(10**9 + 7)))