Linear Sieve Method for Prime Numbers

Problem description:When we calculate for prime numbers with a sieve method,we delete so many numbers which is not necessary repeatly.For instance,there is a number which consists of 3x7x17x23,and we delete it when we delete the multiples of 3 as we delete the same number when we delete the multiples of 7,17,and 23.Please write a program that will not do these jobs more than once.
   
Thinking: There is a factorization theorem:every composite number could be decomposed into the multiplication of some primer numbers.Hence,the number can be decomposed in the form of  (both of p and q are prime numbers and p < q).Therefore,what we need to remove is:,,...and,i=1,2,3.....The value of p and q is the numbers which are not removed currently and in a sequence from small to large.It is easy to write the program.

 

 1 #include <stdio.h>
 2  #define MAX 1000
 3  #define null1 0
 4  #define NEXT(x)  x=next[x]
 5  #define REMOVE(x) {   previous[next[x]]=previous[x];   \
 6                        next[previous[x]]=next[x];       \
 7                    }
 8  
 9  #define INITIAL(n)  { unsigned long i;                    \
10                        for(i=2;i<=n;i++)                   \
11                            previous[i]=i-1,next[i]=i+1;    \
12                        previous[2]=next[n]=null1;           \
13                      }
14  
15  int main()
16  {
17      unsigned long previous[MAX+1]={0};
18      unsigned long next[MAX+1]={0};
19      unsigned long prime,fact,i,mult;
20      unsigned long n;
21      unsigned long count=0;
22      
23      scanf("%lu",&n);
24  
25      INITIAL(n); //initial the array
26  
27      for(prime=2;prime*prime<=n;NEXT(prime))
28      {
29          for(fact=prime;prime*fact<=n;NEXT(fact)) 
30          {
31              for(mult=prime*fact;mult<=n;mult*=prime) 
32                  REMOVE(mult);
33          }
34      }
35      for(i=2;i!=null1;NEXT(i))
36          printf("%lu ",i),count++;
37      printf("\nThe sum of the prime numbers is %lu\n",count);
38  }

Reference material: C语言名题精选百则技巧篇 in Chinese.