Hello Everyone,
Given a number n, print all primes smaller than or equal to n. It is also given that n is a small number.
Example:
Input : n =10
Output : 2 3 5 7Input : n = 20
Output: 2 3 5 7 11 13 17 19
The sieve of Eratosthenes is one of the most efficient ways to find all primes smaller than n when n is smaller than 10 million.
Following is the algorithm to find all the prime numbers less than or equal to a given integer n by the Eratosthene’s method:
When the algorithm terminates, all the numbers in the list that are not marked are prime.
Implementation:
Following is the implementation of the above algorithm. In the following implementation, a boolean array arr of size n is used to mark multiples of prime numbers.
// C++ program to print all primes
// smaller than or equal to
// n using Sieve of Eratosthenes
#include <bits/stdc++.h>
using
namespace
std;
void
SieveOfEratosthenes(
int
n)
{
// Create a boolean array
// "prime[0..n]" and initialize
// all entries it as true.
// A value in prime[i] will
// finally be false if i is
// Not a prime, else true.
bool
prime[n + 1];
memset
(prime,
true
,
sizeof
(prime));
for
(
int
p = 2; p * p <= n; p++)
{
// If prime[p] is not changed,
// then it is a prime
if
(prime[p] ==
true
)
{
// Update all multiples
// of p greater than or
// equal to the square of it
// numbers which are multiple
// of p and are less than p^2
// are already been marked.
for
(
int
i = p * p; i <= n; i += p)
prime[i] =
false
;
}
}
// Print all prime numbers
for
(
int
p = 2; p <= n; p++)
if
(prime[p])
cout << p <<
" "
;
}
// Driver Code
int
main()
{
int
n = 30;
cout <<
"Following are the prime numbers smaller "
<<
" than or equal to "
<< n << endl;
SieveOfEratosthenes(n);
return
0;
}
Output
Following are the prime numbers smaller than or equal to 30 2 3 5 7 11 13 17 19 23 29
Time complexity : O(n*log(log(n)))
*JAVA
// Java program to print all
// primes smaller than or equal to
// n using Sieve of Eratosthenes
class SieveOfEratosthenes {
void sieveOfEratosthenes(int n)
{
// Create a boolean array
// “prime[0…n]” and
// initialize all entries
// it as true. A value in
// prime[i] will finally be
// false if i is Not a
// prime, else true.
boolean prime = new boolean[n + 1];
for (int i = 0; i <= n; i++)
prime[i] = true;
for (int p = 2; p * p <= n; p++)
{
// If prime[p] is not changed, then it is a
// prime
if (prime[p] == true)
{
// Update all multiples of p
for (int i = p * p; i <= n; i += p)
prime[i] = false;
}
}
// Print all prime numbers
for (int i = 2; i <= n; i++)
{
if (prime[i] == true)
System.out.print(i + " ");
}
}
// Driver Code
public static void main(String args[])
{
int n = 30;
System.out.print(
"Following are the prime numbers ");
System.out.println("smaller than or equal to " + n);
SieveOfEratosthenes g = new SieveOfEratosthenes();
g.sieveOfEratosthenes(n);
}
}
*Python
Python program to print all
primes smaller than or equal to
n using Sieve of Eratosthenes
def SieveOfEratosthenes(n):
# Create a boolean array
# "prime[0..n]" and initialize
# all entries it as true.
# A value in prime[i] will
# finally be false if i is
# Not a prime, else true.
prime = [True for i in range(n+1)]
p = 2
while (p * p <= n):
# If prime[p] is not
# changed, then it is a prime
if (prime[p] == True):
# Update all multiples of p
for i in range(p * p, n+1, p):
prime[i] = False
p += 1
# Print all prime numbers
for p in range(2, n+1):
if prime[p]:
print p,
Driver code
if name == ‘main’:
n = 30
print “Following are the prime numbers smaller”,
print “than or equal to”, n
SieveOfEratosthenes(n)