This morning I came across the following little piece of mathematical trivia:

Choose any number (e.g. 14) and write down its divisors:

``````    14
1, 2, 7, 14
``````

Then write down the number of divisors of each of these divisors:

``````    14
1, 2, 7, 14
1, 2, 2, 4
``````

Now the square of the sum of this last group will always equal the sum of its member’s cubes:

``````    (1 + 2 + 2 + 4) ^ 2 = 13 + 23 + 23 + 43
``````

Discovered by Joseph Liouville.

Well, I learned two new things today. Some mathematical trivia and that there was a French Mathematician that I had never heard of called Liouville.

Since I am always on the lookout for simple problems that can work as Python programming exercises, I decided to use the above problem.

Here is my first attempt:

``````def factorise(n):
'''
Given a number return a list of the factors of that number.
'''
factors = 
i = 2
while i <= n:
if n % i == 0:
factors.append(i)
i += 1
return factors

def try_num(n):
factors = factorise(n)
num_factors = []
for factor in factors:
num_factors.append(len(factorise(factor)))

print 'Factors: ', num_factors
print 'Square of sum: ', sum(num_factors) * sum(num_factors)
print 'Sum of cubes: ', sum([factor * factor * factor for factor in num_factors])

def main():
try_num(14)
try_num(144)
try_num(65536)

if __name__ == '__main__':
main()
``````

It works, but while making some small changes, I also noticed that we can do the factorising as a single list comprehension. In other words, we can replace

``````factors = 
i = 2
while i <= n:
if n % i == 0:
factors.append(i)
i += 1
return factors
``````

with

``````return [x + 1 for x in xrange(n) if n % (x + 1) == 0]
``````

Also, we can use a list comprehension for the loop in `try_num`.

This led to the second version of the program:

``````def factorise(n):
'''
Given a number return a list of the factors of that number.
'''
return [x + 1 for x in xrange(n) if n % (x + 1) == 0]

def try_num(n):
num_factors = [len(factorise(factor)) for factor in factorise(n)]

print 'Factors: ', num_factors
print 'Square of sum: ', sum(num_factors) ** 2
print 'Sum of cubes: ', sum([factor ** 3 for factor in num_factors])

def main():
try_num(14)
try_num(144)
try_num(2011)
try_num(65536)

if __name__ == '__main__':
main()
``````

Even after years of using Python, I am still impressed by its ability to express certain things lucidly and compact way.