def sum_primes_below(n):
"""Compute the sum of primes below the given threshold.
>>> sum_primes_below(10)
17
>>> sum_primes_below(2000000)
142913828922
"""
return sum(A3.prime_generator(n))
def find_nth_prime(n):
"""Compute the nth prime number using an unbounded prime generator.
>>> find_nth_prime(6)
13
>>> find_nth_prime(101)
547
>>> find_nth_prime(10001)
104743
"""
primes = A3.prime_generator()
for x in range(n):
prime = primes.next()
return prime
def computed_with_prime_powers(n):
"""A faster method observes that for any prime number, a simple power of
that prime number will be the smallest number with the that factorization
count. For example, 16 is the largest power of 2 less than or equal to 20.
>>> computed_with_prime_powers(20)
232792560
>>> computed_with_prime_powers(40)
5342931457063200
>>> computed_with_prime_powers(80)
32433859254793982911622772305630400L
>>> computed_with_prime_powers(160)
117671955487901874837890815641362681946988303003141220897970719568000L
"""
product = 1
for prime in A3.prime_generator(n):
count = int(math.log(n, prime))
product *= prime ** count
return product
