P3327 [SDOI2015]约数个数和
推导过程
求∑i=1n∑j=1md(ij)\sum_{i = 1} ^{n} \sum_{j = 1} ^{m} d(ij)∑i=1n∑j=1md(ij)
=∑i=1n∑j=1m∑x∣i∑y∣jgcd(x,y)==1= \sum_{i = 1} ^{n} \sum_{j = 1} ^{m} \sum_{x \mid i} \sum_{y \mid j} gcd(x, y) == 1=i=1∑nj=1∑mx∣i∑y∣j∑gcd(x,y)==1
改成枚举x,yx, yx,y,
=∑i=1n∑j=1m∑i∣x∑j∣ygcd(i,j)==1= \sum_{i = 1} ^{n} \sum_{j = 1} ^{m} \sum _{i \mid x} \sum_{j \mid y} gcd(i, j) == 1=i=1∑nj=1∑mi∣x∑j∣y∑gcd(i,j)==1
=∑i=1n∑j=1m⌊ni⌋⌊mj⌋(gcd(i,j)==1)= \sum_{i = 1} ^{n} \sum_{j = 1} ^{m} \lfloor\frac{n}{i}\rfloor \lfloor\frac{m}{j}\rfloor (gcd(i, j) == 1)=i=1∑nj=1∑m⌊in⌋⌊jm⌋(gcd(i,j)==1)
=∑i=1n∑j=1m⌊ni⌋⌊mj⌋∑d∣(gcd(i,j))μ(d)= \sum_{i = 1} ^{n} \sum_{j = 1} ^{m} \lfloor\frac{n}{i}\rfloor \lfloor\frac{m}{j}\rfloor \sum_{d \mid (gcd(i, j))} \mu(d)=i=1∑nj=1∑m⌊in⌋⌊jm⌋d∣(gcd(i,j))∑μ(d)
∑d=1nμ(d)∑i=1nd⌊ndi⌋∑j=1md⌊mdj⌋\sum_{d = 1} ^{n} \mu(d)\sum_{i = 1} ^{\frac{n}{d}} \lfloor\frac{n}{di}\rfloor \sum_{j = 1} ^{\frac{m}{d}} \lfloor \frac{m}{dj}\rfloord=1∑nμ(d)i=1∑dn⌊din⌋j=1∑dm⌊djm⌋
最后我们只要预处理一下所有ndmd\frac{n}{d} \frac{m}{d}dndm的整除分块即可,整体复杂度nnn \sqrt nnn
代码
/*Author : lifehappy
*/
#pragma GCC optimize(2)
#pragma GCC optimize(3)
#include <bits/stdc++.h>#define mp make_pair
#define pb push_back
#define endl '\n'
#define mid (l + r >> 1)
#define lson rt << 1, l, mid
#define rson rt << 1 | 1, mid + 1, r
#define ls rt << 1
#define rs rt << 1 | 1using namespace std;typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> pii;const double pi = acos(-1.0);
const double eps = 1e-7;
const int inf = 0x3f3f3f3f;inline ll read() {ll f = 1, x = 0;char c = getchar();while(c < '0' || c > '9') {if(c == '-') f = -1;c = getchar();}while(c >= '0' && c <= '9') {x = (x << 1) + (x << 3) + (c ^ 48);c = getchar();}return f * x;
}const int N = 5e4 + 10;bool st[N];vector<int> prime;int n, m, mu[N];ll sum[N];void mobius() {st[0] = st[1] = mu[1] = 1;for(int i = 2; i < N; i++) {if(!st[i]) {prime.pb(i);mu[i] = -1;}for(int j = 0; j < prime.size() && i * prime[j] < N; j++) {st[i * prime[j]] = 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];for(ll l = 1, r; l <= i; l = r + 1) {r = i / (i / l);sum[i] += (i / l) * (r - l + 1);}}
}int main() {// freopen("in.txt", "r", stdin);// freopen("out.txt", "w", stdout);// ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);mobius();int T = read();while(T--) {ll n = read(), m = read(), ans = 0;if(n > m) swap(n, m);for(ll l = 1, r; l <= n; l = r + 1) {r = min(n / (n / l), m / (m / l));ans += 1ll * (mu[r] - mu[l - 1]) * sum[n / l] * sum[m / l];}printf("%lld\n", ans);}return 0;
}