Sieve of Sundaram to print all primes smaller than n
Given a number n, print all primes smaller than or equal to n.
Input: n = 10 Output: 2, 3, 5, 7 Input: n = 20 Output: 2, 3, 5, 7, 11, 13, 17, 19
We have discussed Sieve of Eratosthenes algorithm for the above task.
Below is Sieve of Sundaram algorithm.
printPrimes(n) [Prints all prime numbers smaller than n] 1) In general Sieve of Sundaram, produces primes smaller than (2*x + 2) for a number given number x. Since we want primes smaller than n, we reduce n-1 to half. We call it nNew. nNew = (n-1)/2; For example, if n = 102, then nNew = 50. if n = 103, then nNew = 51 2) Create an array marked[n] that is going to be used to separate numbers of the form i+j+2ij from others where 1 <= i <= j 3) Initialize all entries of marked as false. 4) // Mark all numbers of the form i + j + 2ij as true // where 1 <= i <= j Loop for i=1 to nNew a) j = i; b) Loop While (i + j + 2*i*j) 2, then print 2 as first prime. 6) Remaining primes are of the form 2i + 1 where i is index of NOT marked numbers. So print 2i + 1 for all i such that marked[i] is false.
Below is the implementation of above algorithm :
2 3 5 7 11 13 17 19
All red entries in below illustration are marked entries. For every remaining (or black) entry x, the number 2x+1 is prime.
Lets see how it works for n=102, we will have the sieve for (n-1)/2 as follows:
Mark all the numbers which can be represented as i + j + 2ij
Now for all the unmarked numbers in the list, find 2x+1 and that will be the prime:
2*8+1=17 and so on..
How does this work?
When we produce our final output, we produce all integers of the form 2x+1 (i.e., they are odd) except 2 which is handled separately.
Let q be an integer of the form 2x + 1. q is excluded if and only if x is of the form i + j + 2ij. That means, q = 2(i + j + 2ij) + 1 = (2i + 1)(2j + 1) So, an odd integer is excluded from the final list if and only if it has a factorization of the form (2i + 1)(2j + 1) which is to say, if it has a non-trivial odd factor. Source: Wiki
This article is contributed by Anuj Rathore. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready. To complete your preparation from learning a language to DS Algo and many more, please refer Complete Interview Preparation Course.