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#!/usr/bin/env python
# Author: Jared Brothers
# 
# A Python version of the Haskell code from
# "The Genuine Sieve of Eratosthenes"
# www.cs.hmc.edu/~oneill/papers/Sieve-JFP.pdf
#
import heapq, itertools, operator

def sieve(xs):
    """Generate the prime numbers, given an iterable of candidate numbers.
    Cross off multiples of prime numbers incrementally using iterators."""
    table = []
    while 1:
        x = xs.next()
        if table == [] or x < table[0][0]:
            yield x
            xs, ys = itertools.tee(xs)
            timesx = (lambda x: lambda y: x*y)(x)
            heapq.heappush(table, (x**2, itertools.imap(timesx, ys)))
        else:
            while table[0][0] <= x:
                heapq.heapreplace(table, (table[0][1].next(), table[0][1]))

def wheel(factors=[2, 3, 5, 7], next=11):
    """Generate the distances between numbers not divisible by a list of small
    primes, from the next prime up to the product of the list."""
    circumference = reduce(operator.mul, factors)
    prev = next
    next += 1
    end = next + circumference 
    while next < end:
        if not any(next % factor == 0 for factor in factors):
            yield next - prev
            prev = next
        next += 1

def spin(factors=[2, 3, 5, 7], next=11):
    """Generate candidates by making a wheel and cycling through it."""
    for gap in itertools.cycle(wheel(factors, next)):
        yield next
        next += gap

def primes(k=5):
    """Generate primes with the sieve and wheel factorization, which filters
    multiples of the first k primes."""
    smallprimes = list(itertools.islice(sieve(itertools.count(2)), k + 1))
    factors = smallprimes[:-1]
    next = smallprimes[-1]
    return itertools.chain(factors, sieve(spin(factors, next)))