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nth_prime.py
#!/usr/bin/env python
from itertools import islice, count
def eratosthenes_improved(n):
''' Slightly improved version of Eratosthenes Sieve algorithm.
This algorithm do not store an array of numbers to rule out composites.
Instead it uses a dictionary of special composite numbers only, hence saving
memory. It also rules out even numbers in the loop, making less iteration. '''
n = int(n)
# make sure n is an integer
if n < 1: return None
# we're looking for n-th prime!
if n == 1: return 2
# 2 is the only prime that is even
sieve = {}
# here we store our known composite numbers with lowest prime factor
counter = 1
for q in islice(count(3), 0, None, 2):
# we'll only iterate over odd numbers to find a prime
p = sieve.pop(q, None)
if p is None:
# q is a prime, since it is not in known composites
sieve[q*q] = q
# q^2 is a composite with q prime factor
counter += 1
if counter == n:
# we found the n-th prime
break
else:
# Now (N*p)+q is also composite
# let's find the next such composite that is not marked yet
# and mark it with the lowest prime factor p
x = p + q
while x in sieve or not (x&1):
x += p
sieve[x] = p
return q
print eratosthenes_improved(9999) # 104723
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