def problem12():
"""What is the value of the first triangle number to have over five
hundred divisors?"""
# first thing, trinangle numbers can be calculated by the formula:
#
# t(n) = n * (n + 1) / 2
#
# where t(n) is the value of the nth triangle number
#
# after that, we use the Sum of Divisors function [1] to compute
# the number of divisors of a number
from util import primegen
from collections import Counter
from operator import mul
from functools import reduce
def trianglenum(nth):
return nth * (nth + 1) / 2
# we'll reuse our simple `primegen` function to generate primes. but
# we want to save the generated primes, since we'll be using them
# for more than one iteration.
_primes = []
_primes_generator = primegen()
def primes():
for p in _primes:
yield p
for p in _primes_generator:
_primes.append(p)
yield p
# now we go through the triangle numbers, counting their divisors
i = 1
while True:
n = trianglenum(i)
i += 1
factors = []
# break n into prime factors
r = n
g = primes()
while r > 1:
p = next(g)
while r > 1 and r % p == 0:
factors.append(p)
r /= p
# count the prime factors of n
counter = Counter(factors)
# multiply the count of prime factors accordingly to the sum of
# divisors formula
d = reduce(mul, (n + 1 for n in counter.values()), 1)
if d > 500:
return n
def problem10():
"""Find the sum of all the primes below two million."""
from itertools import takewhile
ceil = 2000000
return sum(n for n in takewhile(lambda x: x < ceil, util.primegen()))