Number theory-Euler function explanation and template

Euler function: For any positive integer N, the number of positive integers (including 1) that are less than or equal to N ([1,N]) and relatively prime to N is called the Euler function to N, which is denoted as φ(n). Note: Coprime means that the two have no common factor other than 1.

It is stipulated that φ(1)=1.

The properties of Euler's function:

1. If p is a prime number, then

2. If a is a prime number and a|x, then

2. If a and b are relatively prime, then

Find the Euler function value of a number:

Decompose a number into prime factors:

It can be seen from property 1

         

So the code is as follows:

int phi(int n) {
	int res=n;
	for(int i=2;i*i<=n;++i) {
		if(n%i==0) res=res/i*(i-1);
		while(n%i==0) n/=i;
	}
	if(n>1) res=res/n*(n-1);
	return res;
} 

Combined with the sieve method

int phi[maxn];
void init(int n) {
	for(int i=1;i<=n;++i) phi[i]=i;
	for(int i=2;i<=n;i++) {
		if(phi[i]==i){
			for(int j=i;j<=n;j+=i) phi[j]=phi[j]/i*(i-1);
		} 
	}
}

Combine with Euler sieve

int phi[maxn];
int prime[maxn],cnt;
int isnp[maxn];
void init(int n) {
	phi[1]=1; cnt=0;
	for(int i=2;i<=n;i++) 
		if(!isnp[i])
			prime[++cnt]=i,phi[i]=i-1;
		for(int j=1;j<=cnt&&i*prime[j]<=n;j++){
			isnp[i*prime[j]]=1;
			if(i%prime[j]==0) {
				phi[prime[j]*i]=phi[i]*prime[j]; 
				break;
			} 
			else phi[prime[j]*i]=phi[i]*phi[prime[j]];
		}
			
}