做网站文件夹都起什么名字,商城网站建设实例需求,dnf制裁做任务网站,网站建设管理制度1237 最大公约数之和 V3
推式子 ∑i1n∑j1ngcd(i,j)∑d1nd∑i1n∑j1n(gcd(i,j)d)∑d1nd∑i1nd∑j1nd(gcd(i,j)1)∑d1nd∑i1nd∑j1nd∑k∣gcd(i,j)μ(k)∑d1nd∑k1ndμ(k)∑i1nkd∑j1nkd1套路地设tkd∑t1n(⌊nt⌋)2∑d∣tdμ(td)∑t1n(⌊nt⌋)2ϕ(t)接下来就是杜教筛求∑i1nϕ(…1237 最大公约数之和 V3
推式子
∑i1n∑j1ngcd(i,j)∑d1nd∑i1n∑j1n(gcd(i,j)d)∑d1nd∑i1nd∑j1nd(gcd(i,j)1)∑d1nd∑i1nd∑j1nd∑k∣gcd(i,j)μ(k)∑d1nd∑k1ndμ(k)∑i1nkd∑j1nkd1套路地设tkd∑t1n(⌊nt⌋)2∑d∣tdμ(td)∑t1n(⌊nt⌋)2ϕ(t)接下来就是杜教筛求∑i1nϕ(i)了那这不就是杜教筛水题了嘛。\sum_{i 1} ^{n} \sum_{j 1} ^{n} gcd(i, j)\\ \sum_{d 1} ^{n} d\sum_{i 1} ^ {n} \sum_{j 1} ^ {n} (gcd(i, j) d)\\ \sum_{d 1} ^{n} d\sum_{i 1} ^{\frac{n}{d}} \sum_{j 1} ^{\frac{n}{d}}(gcd(i, j) 1)\\ \sum_{d 1} ^{n} d\sum_{i 1} ^{\frac{n}{d}} \sum_{j 1} ^{\frac{n}{d}} \sum_{k \mid gcd(i, j)} \mu(k)\\ \sum_{d 1} ^{n} d\sum_{k 1} ^{\frac{n}{d}} \mu(k) \sum_{i 1} ^{\frac{n}{kd}} \sum_{j 1} ^{\frac{n}{kd}}1\\ 套路地设t kd\\ \sum_{t 1} ^{n} \left(\lfloor\frac{n}{t}\rfloor \right) ^ 2 \sum_{d \mid t} d \mu(\frac{t}{d})\\ \sum_{t 1} ^{n} \left(\lfloor\frac{n}{t}\rfloor \right) ^ 2 \phi(t)\\ 接下来就是杜教筛求\sum_{i 1} ^{n} \phi(i)了那这不就是杜教筛水题了嘛。 i1∑nj1∑ngcd(i,j)d1∑ndi1∑nj1∑n(gcd(i,j)d)d1∑ndi1∑dnj1∑dn(gcd(i,j)1)d1∑ndi1∑dnj1∑dnk∣gcd(i,j)∑μ(k)d1∑ndk1∑dnμ(k)i1∑kdnj1∑kdn1套路地设tkdt1∑n(⌊tn⌋)2d∣t∑dμ(dt)t1∑n(⌊tn⌋)2ϕ(t)接下来就是杜教筛求i1∑nϕ(i)了那这不就是杜教筛水题了嘛。
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 8e6 10, mod 1000000007;ll phi[N], inv2;int prime[N], cnt;bool st[N];ll quick_pow(ll a, ll n, ll mod) {ll ans 1;while(n) {if(n 1) ans ans * a % mod;a a * a % mod;n 1;}return ans;
}void init() {phi[1] 1;for(int i 2; i N; i) {if(!st[i]) {prime[cnt] i;phi[i] i - 1;}for(int j 0; j cnt 1ll * 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] phi[i]) % mod;}inv2 quick_pow(2, mod - 2, mod);
}ll calc(ll x) {x % mod;return x * (x 1) % mod * inv2 % mod;
}mapll, ll ans_phi;ll get_phi(ll x) {if(x N) return phi[x];if(ans_phi.count(x)) return ans_phi[x];ll ans calc(x);for(ll l 2, r; l x; l r 1) {r x / (x / l);ans (ans - (r - l 1) % mod * get_phi(x / l) % mod mod) % mod;}return ans_phi[x] ans;
}ll calc2(ll x) {x % mod;return x * x % mod;
}int main() {// freopen(in.txt, r, stdin);// freopen(out.txt, w, stdout);// ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);ll n read(), ans 0;init();for(ll l 1, r; l n; l r 1) {r n / (n / l);ans (ans calc2(n / l) * (get_phi(r) - get_phi(l - 1)) % mod mod) % mod;}cout ans endl;return 0;
}
