P6810 「MCOI-02」Convex Hull 凸包 题解
题目链接
看绝大多数题解都是莫反，这里写个不用莫反的（其实就是推式子）
∑ i = 1 n ∑ j = 1 m τ ( i ) τ ( j ) τ ( gcd ( i , j ) ) \sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}\tau(i)\tau(j)\tau(\gcd(i,j)) i=1∑nj=1∑mτ(i)τ(j)τ(gcd(i,j))
∑ i = 1 n τ ( i ) ∑ j = 1 m τ ( j ) τ ( gcd ( i , j ) ) \sum\limits_{i=1}^{n}\tau(i)\sum\limits_{j=1}^{m}\tau(j)\tau(\gcd(i,j)) i=1∑nτ(i)j=1∑mτ(j)τ(gcd(i,j))
∑ i = 1 n τ ( i ) ∑ j = 1 m τ ( j ) ∑ k ∣ i , k ∣ j 1 \sum\limits_{i=1}^{n}\tau(i)\sum\limits_{j=1}^{m}\tau(j)\sum\limits_{k|i,k|j}1 i=1∑nτ(i)j=1∑mτ(j)k∣i,k∣j∑1
接下来优先枚举 k k k
∑ k = 1 min ( n , m ) ∑ k ∣ i , i ≤ n τ ( i ) ∑ k ∣ j , j ≤ m τ ( j ) \sum\limits_{k=1}^{\min(n,m)}\sum\limits_{k|i,i\le n}\tau(i)\sum\limits_{k|j,j \le m}\tau(j) k=1∑min(n,m)k∣i,i≤n∑τ(i)k∣j,j≤m∑τ(j)
设 s k = ∑ k ∣ i , i ≤ n τ ( i ) s_k=\sum\limits_{k|i,i\le n}\tau(i) sk=k∣i,i≤n∑τ(i)， c k = ∑ k ∣ i , i ≤ m τ ( i ) c_k=\sum\limits_{k|i,i\le m}\tau(i) ck=k∣i,i≤m∑τ(i)
那么原式等于 ∑ k = 1 min ( n , m ) s k ⋅ c k \sum\limits_{k=1}^{\min(n,m)}s_k \cdot c_k k=1∑min(n,m)sk⋅ck
我们可以 O ( n log n ) O(n \log n) O(nlogn) 预处理出 τ ( i ) , s i , c i \tau(i),s_i,c_i τ(i),si,ci，代入上式即可
#include