建网站有哪些费用,沐众科技网站建设,电子邮箱怎么注册,网站托管平台P1447 [NOI2010]能量采集
式子化简
显然题目就是要我们求∑i1n∑j1m2gcd(i,j)−1\sum_{i 1} ^{n} \sum_{j 1} ^{m} 2gcd(i, j) - 1∑i1n∑j1m2gcd(i,j)−1 2∑i1n∑j1mgcd(i,j)−nm 2\sum_{i 1} ^{n} \sum_{j 1} ^{m} gcd(i, j) - nm2i1∑nj1∑mgcd(i,j)−nm
转…P1447 [NOI2010]能量采集
式子化简
显然题目就是要我们求∑i1n∑j1m2gcd(i,j)−1\sum_{i 1} ^{n} \sum_{j 1} ^{m} 2gcd(i, j) - 1∑i1n∑j1m2gcd(i,j)−1
2∑i1n∑j1mgcd(i,j)−nm 2\sum_{i 1} ^{n} \sum_{j 1} ^{m} gcd(i, j) - nm2i1∑nj1∑mgcd(i,j)−nm
转化为我们要求∑i1n∑j1mgcd(i,j)\sum_{i 1} ^{n} \sum_{j 1} ^{m} gcd(i, j)∑i1n∑j1mgcd(i,j)
∑d1nd∑i1nd∑j1mdgcd(i,j)1 \sum_{d 1} ^{n}d\sum_{i 1} ^{\frac{n}{d}} \sum_{j 1} ^{\frac{m}{d}} gcd(i, j) 1d1∑ndi1∑dnj1∑dmgcd(i,j)1
套上mobiusmobiusmobius
∑d1nd∑i1nd∑j1md∑k∣gcd(i,j)μ(k) \sum_{d 1} ^{n}d\sum_{i 1} ^{\frac{n}{d}} \sum_{j 1} ^{\frac{m}{d}} \sum_{k \mid gcd(i, j)} \mu(k)d1∑ndi1∑dnj1∑dmk∣gcd(i,j)∑μ(k)
∑d1nd∑i1nd∑j1md∑k∣gcd(i,j)μ(k) \sum_{d 1} ^{n}d\sum_{i 1} ^{\frac{n}{d}} \sum_{j 1} ^{\frac{m}{d}} \sum_{k \mid gcd(i, j)} \mu(k)d1∑ndi1∑dnj1∑dmk∣gcd(i,j)∑μ(k)
∑d1nd∑k1ndμ(k)⌊ndk⌋⌊mdk⌋ \sum_{d 1} ^{n} d\sum_{k 1} ^{\frac{n}{d}}\mu(k) \lfloor\frac{n}{dk}\rfloor \lfloor\frac{m}{dk}\rfloord1∑ndk1∑dnμ(k)⌊dkn⌋⌊dkm⌋
另tdkt dktdk
∑t1n⌊nt⌋⌊mt⌋∑d∣tdμ(td) \sum_{t 1} ^{n} \lfloor\frac{n}{t}\rfloor \lfloor\frac{m}{t}\rfloor \sum_{d \mid t}d\mu(\frac{t}{d})t1∑n⌊tn⌋⌊tm⌋d∣t∑dμ(dt)
mobiusmobiusmobius反演有∑d∣nμ(d)dϕ(n)n\sum_{d\mid n}\frac{\mu(d)}{d} \frac{\phi(n)}{n}∑d∣ndμ(d)nϕ(n)
∑t1n⌊nt⌋⌊mt⌋ϕ(t) \sum_{t 1} ^{n} \lfloor\frac{n}{t}\rfloor \lfloor\frac{m}{t}\rfloor \phi(t)t1∑n⌊tn⌋⌊tm⌋ϕ(t)
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 pairint, 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 1e7 10;bool st[N];vectorint prime;int n, m;ll phi[N];void mobius() {st[0] st[1] phi[1] 1;for(int i 2; i N; i) {if(!st[i]) {prime.pb(i);phi[i] i - 1;}for(int j 0; j prime.size() i * prime[j] N; j) {st[i * prime[j]] 1;if(i % prime[j] 0) {phi[i * prime[j]] phi[i] * prime[j];break;}phi[i * prime[j]] phi[i] * (prime[j] - 1);}}for(int i 1; i N; i) phi[i] phi[i - 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();ll n read(), m read();if(n m) swap(n, m);ll ans 0;for(ll l 1, r; l n; l r 1) {r min(n / (n / l), m / (m / l));ans (n / l) * (m / l) * (phi[r] - phi[l - 1]);}printf(%lld\n, 2 * ans - n * m);return 0;
}
