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Sieve of Eratosthenes

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Sieve of Eratosthenes

Topic: Sieve of Eratosthenes

In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking as composite the multiples of each prime, starting with the first prime number, 2.

Example:-

Input : n = 10
Output : 2 3 5 7 

Input : n = 20 
Output: 2 3 5 7 11 13 17 19

Steps to Implement sieve of Eratosthenes

  • Create a list of consecutive integers from 2 to n: (2, 3, 4, …, n).

  • Initially, let p equal 2, the first prime number.

  • Starting from p2, count up in increments of p and mark each of these numbers greater than or equal to p2 itself in the list. These numbers will be p(p+1), p(p+2), p(p+3), etc..

  • Find the first number greater than p in the list that is not marked. If there was no such number, stop. Otherwise, let p now equal this number (which is the next prime), and repeat from step 3.

Implementation of Sieve of Eratosthenes in Python3.

def get_primes(n):
  m = n+1
  #numbers = [True for i in range(m)]
  numbers = [True] * m #EDIT: faster
  for i in range(2, int(n**0.5 + 1)):
    if numbers[i]:
      for j in range(i*i, m, i):
        numbers[j] = False
  primes = []
  for i in range(2, m):
    if numbers[i]:
      primes.append(i)
  return primes

print(get_primes(25))

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