Given a number n, print all primes smaller than or equal to n. It is also given that n is a small number.
Input : n =10 Output : 2 3 5 7 Input : 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 or so (Ref Wiki).
Following is the algorithm to find all the prime numbers less than or equal to a given integer n by Eratosthenes’ method:
- 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.
When the algorithm terminates, all the numbers in the list that are not marked are prime.
Explanation with Example:
Let us take an example when n = 50. So we need to print all print numbers smaller than or equal to 50.
Now we move to our next unmarked number 3 and mark all the numbers which are multiples of 3 and are greater than or equal to the square of it.
So the prime numbers are the unmarked ones: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.
Thanks to Krishan Kumar for providing above explanation.
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.
Following are the prime numbers below 30 2 3 5 7 11 13 17 19 23 29
Time complexity : O(n*log(log(n)))
This article is compiled by Abhinav Priyadarshi and reviewed by GeeksforGeeks team. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
- Sieve of Eratosthenes in 0(n) time complexity
- Sum of all Primes in a given range using Sieve of Eratosthenes
- Bitwise Sieve
- Segmented Sieve
- Sieve of Atkin
- Sieve of Sundaram to print all primes smaller than n
- Segmented Sieve (Print Primes in a Range)
- Number of unmarked integers in a special sieve
- Prime Factorization using Sieve O(log n) for multiple queries
- Count of integers from the range [0, N] whose digit sum is a multiple of K
- Find the sum of the costs of all possible arrangements of the cells
- Find an integer in the given range that satisfies the given conditions
- Maximum sum subarray after altering the array
- Minimum cost to partition the given binary string