[Template] Linear Sieve Prime Number (Eulerian Sieve Method)

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topic

Luogu P3383 Linear Sieve Prime Number

answer

The essence of sieve method is to mark composite numbers, that is, to mark all multiples of each number. However, when marking, a composite number will be marked multiple times by all its prime factors, resulting in a waste of time. Therefore, consider that each composite number is only marked by its own smallest prime factor.

Algorithm flow: Use v[i] to represent the smallest prime factor of i, use all prime numbers p not greater than v[i] to mark the composite number i*p, and update v[i*p]=p.

The number of prime numbers between 1 and n is approximately  .

Code

#include <bits/stdc++.h>
using namespace std;
typedef long long ll;
const int maxn=1e8+5;
const int maxp=5e6;
int n,m,q;
int v[maxn],prime[maxp];
inline void get_prime()
{
	for(int i=2;i<=n;i++)	//注意从2开始循环 
	{
		if(!v[i]) {v[i]=i; prime[++m]=i;}
		for(int j=1;j<=m;j++)
		{
			
			if(prime[j]>v[i]||prime[j]*i>n) break;
			v[i*prime[j]]=prime[j];
		}
	}
}

int main()
{
	scanf("%d%d",&n,&q);
	get_prime();
	for(int i=1;i<=q;i++) 
	{
		int k; scanf("%d",&k);
		printf("%d\n",prime[k]);
	}
	
	return 0;
}