欧拉计划题目中的一些非常精致的解法

problem 47

The first two consecutive numbers to have two distinct prime factors are:

14 = 2  7
15 = 3  5

The first three consecutive numbers to have three distinct prime factors are:

644 = 2²  7  23
645 = 3  5  43
646 = 2  17  19.

Find the first four consecutive integers to have four distinct prime factors. What is the first of these numbers?

这里有一个非常精致的解（在论坛上看到的）

lim = 1000000
l = 4
sieve = [0] * lim
ls = [l] * l

for i in range(2,int(math.sqrt(lim)) + 1):
    if sieve[i] == 0:
        sieve[i+i::i] = [x + 1 for x in sieve[i+i::i]]

for i in range(2,len(sieve) - l):
    if sieve[i:i+l] == ls:
        print i
        break

该解法将2到1000000中的每个数，统计其所含有的素数因子的个数，若有连续4个数含有4个素数因子，既符合题目条件。该方法也可以用于获得素数列表，如下所示

def primes(limit):
    a = [True] * limit                          
    a[0] = a[1] = False
    for (i, isprime) in enumerate(a):
        if isprime:
            yield i
            for n in range(i*i, limit, i):     
                a[n] = False