# Problem

Given an integer N, generate all primes less than or equal to N.

# Sieve of Eratosthenes – Explanation

Sieve of Eratosthenes is an algorithm that generates all prime up to N. Read this article written by Jane Alam Jan on Generating Primes – LightOJ Tutorial. The pdf contains almost all the optimizations of the Sieve of Eratosthenes.

# Code

vector <int> prime; // Stores generated primes
char sieve[SIZE]; // 0 means prime
void primeSieve ( int n ) {
sieve[0] = sieve[1] = 1; // 0 and 1 are not prime

prime.push_back(2); // Only Even Prime
for ( int i = 4; i <= n; i += 2 ) sieve[i] = 1; // Remove multiples of 2

int sqrtn = sqrt ( n );
for ( int i = 3; i <= sqrtn; i += 2 ) {
if ( sieve[i] == 0 ) {
for ( int j = i * i; j <= n; j += 2 * i ) sieve[j] = 1;
}
}

for ( int i = 3; i <= n; i += 2 ) if ( sieve[i] == 0 ) prime.push_back(i);
}


Some lines from the code above can be omitted depending on the situation. But as a whole, the above code gives us two products, a vector<int> prime which contains all generated primes and a char[] sieve that indicates whether a particular value is prime or not. We will need the sieve array for optimizations in other algorithms.

# Complexity

The algorithm has a runtime complexity of $O(N log (logN ) )$