Sol
推导:(n<m,p为质数)
(ans=sum_psum_{i=1}^{frac{n}{p}}mu(i)frac{n}{pi}frac{m}{pi})
(=sum_{k=1}^{n}frac{n}{k}frac{m}{k}sum_{p|k}mu(frac{k}{p}))
(sum_{p|k}mu(frac{k}{p}))可以暴力预处理,也可以在筛的时候计算出
暴力求
# include <bits/stdc++.h>
# define RG register
# define IL inline
# define Zsydalao 666
# define Fill(a, b) memset(a, b, sizeof(a))
using namespace std;
typedef long long ll;
const int _(1e7 + 1);
IL ll Read(){
char c = '%'; ll x = 0, z = 1;
for(; c > '9' || c < '0'; c = getchar()) if(c == '-') z = -1;
for(; c >= '0' && c <= '9'; c = getchar()) x = x * 10 + c - '0';
return x * z;
}
int prime[_], num, mu[_], f[_];
bool isprime[_];
IL void Prepare(){
isprime[1] = 1; mu[1] = 1;
for(RG int i = 2; i < _; ++i){
if(!isprime[i]) prime[++num] = i, mu[i] = -1;
for(RG int j = 1; j <= num && i * prime[j] < _; ++j){
isprime[i * prime[j]] = 1;
if(i % prime[j]) mu[i * prime[j]] = -mu[i];
else{ mu[i * prime[j]] = 0; break; }
}
}
for(RG int i = 1; i < _; ++i)
for(RG int j = 1; j <= num && i * prime[j] < _; ++j)
f[i * prime[j]] += mu[i];
for(RG int i = 1; i < _; ++i) f[i] += f[i - 1];
}
int main(RG int argc, RG char *argv[]){
Prepare();
for(RG int T = Read(); T; --T){
RG ll n = Read(), m = Read(), ans = 0;
if(n > m) swap(n, m);
for(RG ll k = 1, j; k <= n; k = j + 1){
j = min(n / (n / k), m / (m / k));
ans += (n / k) * (m / k) * (f[j] - f[k - 1]);
}
printf("%lld
", ans);
}
return 0;
}
筛的时候处理
# include <bits/stdc++.h>
# define RG register
# define IL inline
# define Zsydalao 666
# define Fill(a, b) memset(a, b, sizeof(a))
using namespace std;
typedef long long ll;
const int _(1e7 + 1);
IL ll Read(){
char c = '%'; ll x = 0, z = 1;
for(; c > '9' || c < '0'; c = getchar()) if(c == '-') z = -1;
for(; c >= '0' && c <= '9'; c = getchar()) x = x * 10 + c - '0';
return x * z;
}
int prime[_], num, mu[_], f[_];
bool isprime[_];
IL void Prepare(){
isprime[1] = 1; mu[1] = 1;
for(RG int i = 2; i < _; ++i){
if(!isprime[i]) prime[++num] = i, mu[i] = -1, f[i] = 1;
for(RG int j = 1; j <= num && i * prime[j] < _; ++j){
isprime[i * prime[j]] = 1;
if(i % prime[j]) mu[i * prime[j]] = -mu[i], f[i * prime[j]] = mu[i] - f[i];
else{ mu[i * prime[j]] = 0; f[i * prime[j]] = mu[i]; break; }
}
}
for(RG int i = 1; i < _; ++i) f[i] += f[i - 1];
}
int main(RG int argc, RG char *argv[]){
Prepare();
for(RG int T = Read(); T; --T){
RG ll n = Read(), m = Read(), ans = 0;
if(n > m) swap(n, m);
for(RG ll k = 1, j; k <= n; k = j + 1){
j = min(n / (n / k), m / (m / k));
ans += (n / k) * (m / k) * (f[j] - f[k - 1]);
}
printf("%lld
", ans);
}
return 0;
}