【模板】数论板子

数论分块

用于求解

\[\sum\limits_{i=1}^{n}f_i\cdot \left\lfloor\dfrac{n}{i}\right\rfloor \]

亦可求解多维

\[\sum\limits_{i=1}^{\min(n_1,n_2,\cdots,n_k)}(f_i\cdot \prod\limits_{j=1}^{k}\left\lfloor\dfrac{n_j}{i}\right\rfloor) \]

前提是求出了数论函数\(f(n)\)的前缀和。

ull NTSD(ull pre[],int n) {
	//number-theory;
	//sqrt-decomposition;
	int l = 1, r = 0;
	ull result = 0;
	while(l <= n) {
		r = (int)floor(n/floor(1.0*n/l));
		result += (pre[r]-pre[l-1])*1ll*floor(n/l);
		l = r+1;
	}
	return result;
}
ull NTSDQ(ull pre[],int n,int m) {
	int l = 1, r = 0;
	ull result = 0;
	while(l <= min(m,n)) {
		r = (int)min(n/(1.0*n/l),m/(1.0*m/l));
		result += (pre[r]-pre[l-1])*1ll*(n/l)*(m/l);
		l = r+1;
	}
	return result;
}

筛法

\(prime,\varphi(n),\mu(n)\)

int mu[z];
bitset<z> b;
ull prime[z];
ull phi[z];
ull minp[z];
void line_prime(int n) {
	b.reset();
	for(int i = 2;i <= n;++i) {
		if(!b[i]) 
			prime[++prime[0]] = i;
		for(int j = 1;j <= prime[0];++j) {
			if(i*prime[j] > n) break;
			if(!minp[i*prime[j]]) 
				minp[i*prime[j]] = prime[j];
			b[i*prime[j]] = 1;
			if(i%prime[j] == 0) 
				break;
		}
	}
}
void line_phi(int n) {
	b.reset();
	phi[1] = 1;
	for(int i = 2;i <= n;++i) {
		if(!b[i]) {
			prime[++prime[0]] = i;
			phi[i] = i-1;
		}
		for(int j = 1;j <= prime[0];++j) {
			if(i*prime[j] > n) break;
			if(i%prime[j]) 
				phi[i*prime[j]] = phi[i]*phi[prime[j]];
			else {
				phi[i*prime[j]] = phi[i]*prime[j];
				break;
			}
		}
	}
}
void line_mu(int n) {
	b.reset();
	mu[1] = 1;
	for(int i = 2;i <= n;++i) {
		if(!b[i]) {
			mu[i] = -1;
			prime[++prime[0]] = i;
		}
		for(int j = 1;j <= prime[0];++j) {
			if(i*j > n) break;
			b[i*prime[j]] = 1;
			if(i%prime[j] == 0) {
				mu[i*prime[j]] = 0;
				break;
			}
			mu[i*prime[j]] = -mu[i];
		}
	}
}
ull tau[z], sigma;
ull num[z];
void line_tau(int n) {
	tau[1] = 1;
	for(int i = 2;i <= n;++i) {
		if(!b[i]) {
			b[i] = 1;
			prime[++prime[0]] = i;
			tau[i] = 2;
			num[i] = 1;
		}
		for(int j = 1;j <= prime[0];++j) {
			if(i*prime[j] > n) break;
			b[i*prime[j]] = 1;
			if(i%prime[j] == 0) {
				num[i*prime[j]] = num[i]+1;
				tau[i*prime[j]] = tau[i]/num[i*prime[j]]*(num[i*prime[j]]+1);
				break;
			} else {
				num[i*prime[j]] = 1;
				tau[i*prime[j]] = tau[i]*2;
			}
		}
	}
}

标签：prime,tau,phi,数论,板子,break,int,ull,模板
来源： https://www.cnblogs.com/bikuhiku/p/NTep.html