先利用差分
思想，求(f(b,d,k) - f(a - 1,d,k) - f(b,c-1,k)+f(a-1,c-1,k))
[egin{aligned}
f(n,m,k) &=sum_{i=1}^{n} sum_{j=1}^{m} [gcd(i,j) = k]\
&= sum_{i=1}^{frac nk} sum_{j=1}^{frac mk} [gcd(i,j) = 1]\
end{aligned}\
]
然后莫比乌斯反演
[sum_{i=1}^{n} sum_{j=1}^{m} [gcd(i,j) = 1]\
sum_{i=1}^{n} sum_{j=1}^{m} sum_{p|i,p|j}mu(p)\
sum_{p=1}^{n}mu(p)sum_{i=1}^{frac np}sum_{j=1}^{frac mp}1\
sum_{p=1}^{n}mu(p)(frac np)(frac mp)\
]
最后数论分块。
code
#include<bits/stdc++.h>
using namespace std;
int p[50050],prime[50050],cnt,mu[50050];
void init(int N){
p[0] = p[1] = 1; mu[1] = 1;
for(int i = 2; i <= N; ++ i){
if(!p[i]) { prime[++ cnt] = i; mu[i] = -1; }
for(int j = 1; 1ll * prime[j] * i <= N; ++ j){
p[prime[j] * i] = 1;
if(i % prime[j] == 0) break;
mu[i * prime[j]] = -mu[i];
}
}
for(int i = 1; i <= N; ++ i) mu[i] += mu[i - 1];
}
int f(int n, int m, int k){
if(n > m) swap(n,m);
n /= k; m /= k;
if(n == 0 || m == 0) return 0;
int ret = 0;
for(int l = 1, r; l <= n; l = r + 1){
r = min(min(n/(n/l), m/(m/l)), n);
ret = ret + 1ll * (mu[r] - mu[l - 1]) * (n/l) * (m/l);
}
return ret;
}
int main(){
init(50000);
int T; scanf("%d",&T);
while(T --){
int a,b,c,d,k;
scanf("%d%d%d%d%d",&a,&b,&c,&d,&k);
printf("%d
",f(b,d,k) - f(a - 1,d,k) - f(b,c - 1,k) + f(a - 1,c - 1,k));
}
return 0;
}