JavaAlgorithms/Others/SieveOfEratosthenes.java

package Others;

import java.util.Arrays;

/**
 * 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 (i.e., not prime) the multiples of each prime,
 * starting with the first prime number, 2.
 * The multiples of a given prime are generated as a sequence of numbers starting from that prime,
 * with constant difference between them that is equal to that prime.
 * This is the sieve's key distinction from using trial division to sequentially test each
 * candidate number for divisibility by each prime.
 * Once all the multiples of each discovered prime have been marked as composites, the remaining
 * unmarked numbers are primes.
 * <p>
 * Poetry about Sieve of Eratosthenes:
 *    <p><i>Sift the Two's and Sift the Three's:</i></p>
 *    <p><i>The Sieve of Eratosthenes.</i></p>
 *    <p><i>When the multiples sublime,</i></p>
 *    <p><i>The numbers that remain are Prime.</i></p>
 *
 * @see <a href="https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes">Wiki</a>
 */
public class SieveOfEratosthenes {

  /**
   * @param n The number till which we have to check for prime Prints all the prime numbers till n.
   *          Should be more than 1.
   * @return array of all prime numbers between 0 to n
   */
  public static int[] findPrimesTill(int n) {
    // Create array where index is number and value is flag - is that number a prime or not.
    // size of array is n + 1 cause in Java array indexes starts with 0
    Type[] numbers = new Type[n + 1];

    // Start with assumption that all numbers except 0 and 1 are primes.
    Arrays.fill(numbers, Type.PRIME);
    numbers[0] = numbers[1] = Type.NOT_PRIME;

    double cap = Math.sqrt(n);
    // Main algorithm: mark all numbers which are multiples of some other values as not prime
    for (int i = 2; i <= cap; i++) {
      if (numbers[i] == Type.PRIME) {
        for (int j = 2; i * j <= n; j++) {
          numbers[i * j] = Type.NOT_PRIME;
        }
      }
    }

    //Write all primes to result array
    int primesCount = (int) Arrays.stream(numbers)
        .filter(element -> element == Type.PRIME)
        .count();
    int[] primes = new int[primesCount];

    int primeIndex = 0;
    for (int i = 0; i < n + 1; i++) {
      if(numbers[i] == Type.PRIME) {
        primes[primeIndex++] = i;
      }
    }

    return primes;
  }

  private enum Type {
    PRIME, NOT_PRIME
  }

  public static void main(String[] args) {
    int n = 100;
    System.out.println("Searching for all primes from zero to " + n);
    int[] primes = findPrimesTill(n);
    System.out.println("Found: " + Arrays.toString(primes));
  }
}