The sequence of triangle numbers is generated by adding the natural numbers. So the 7th triangle number would be 1 + 2 + 3 + 4 + 5 + 6 + 7 = 28. The first ten terms would be:

1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...

Let us list the factors of the first seven triangle numbers:

1: 1

3: 1,3

6: 1,2,3,6

10: 1,2,5,10

15: 1,3,5,15

21: 1,3,7,21

28: 1,2,4,7,14,28

We can see that 28 is the first triangle number to have over five divisors.

What is the value of the first triangle number to have over five hundred divisors?

In [17]:

def tri_num(num):
    tri = 0
    
    for x in xrange(1, num):
        tri += x
        yield tri

In [23]:

tri_list = [i for i in tri_num(10000)]

In [24]:

def find_factors(lst):
    
    for i in xrange(len(lst)):
        num_factors = 0
        for x in xrange(1, lst[i]):
            if lst[i]%x == 0:
                num_factors += 1
                if num_factors >= 500:
                    return lst[i]
                    
    print "range too small"

In [25]:

find_factors(tri_list)

---------------------------------------------------------------------------
KeyboardInterrupt                         Traceback (most recent call last)
<ipython-input-25-68c81c2995eb> in <module>()
----> 1 find_factors(tri_list)

<ipython-input-24-ba9584544721> in find_factors(lst)
      4         num_factors = 0
      5         for x in xrange(1, lst[i]):
----> 6             if lst[i]%x == 0:
      7                 num_factors += 1
      8                 if num_factors >= 500:

KeyboardInterrupt:

The above method is too slow. A more efficient method is needed.

In [28]:

import math

def factors2(num):
    divisors = 0
    square = int(math.sqrt(num))
    
    for i in xrange(1, square):
        if num % i == 0:
            divisors += 2
            
    if square * square == num:
        divisors -= 1
        
    return divisors

In [31]:

num = 0
x = 1

while(factors2(num) < 500):
    num += x
    x += 1
    
print num    

76576500

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