Question

This is my code in python for calculation of sum of prime numbers less than a given number.
What more can I do to optimize it?

import math
primes = [2,]                      #primes store the prime numbers



for i in xrange(3,20000,2):                    #i is the test number
    x = math.sqrt(i)
    isprime = True
    for j in primes:               #j is the devider. only primes are used as deviders
        if j <= x:
            if i%j == 0:
                    isprime = False
                    break


    if isprime:
        primes.append(i,)


print sum (primes,)

Answer 1

You can use a different algorithm called the Sieve of Eratosthenes which will be faster but take more memory. Keep an array of flags, signifying whether each number is a prime or not, and for each new prime set it to zero for all multiples of that prime.

N = 10000

# initialize an array of flags
is_prime = [1 for num in xrange(N)]
is_prime[0] = 0 # this is because indexing starts at zero
is_prime[1] = 0 # one is not a prime, but don't mark all of its multiples!

def set_prime(num):
    "num is a prime; set all of its multiples in is_prime to zero"
    for x in xrange(num*2, N, num):
        is_prime[x] = 0

# iterate over all integers up to N and update the is_prime array accordingly
for num in xrange(N):
    if is_prime[num] == 1:
        set_prime(num)

primes = [num for num in xrange(N) if is_prime[num]]

You can actually do this for pretty large N if you use an efficient bit array, such as in this example (scroll down on the page and you’ll find a Sieve of Eratosthenes example).