G. GCD Festival
∑i=1n∑j=1ngcd(ai,aj)gcd(i,j)∑d=1nd∑i=1nd∑j=1ndgcd(aid,ajd)[gcd(i,j)=1]∑d=1nd∑k=1ndμ(k)∑i=1nkd∑j=1nkdgcd(aikd,ajkd)T=kd∑T=1n∑i=1nT∑j=1nTgcd(aiT,ajT)∑d∣Tdμ(Td)∑T=1nϕ(T)∑i=1nT∑j=1nTgcd(aiT,ajT)\sum_{i = 1} ^{n} \sum_{j = 1} ^{n} \gcd(a_i, a_j) \gcd(i, j)\\ \sum_{d = 1} ^{n} d \sum_{i = 1} ^{\frac{n}{d}} \sum_{j = 1} ^{\frac{n}{d}} \gcd(a_{id}, a_{jd})[\gcd(i, j) = 1]\\ \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}} \gcd(a_{i kd}, a_{jkd})\\ T = kd\\ \sum_{T = 1} ^{n} \sum_{i = 1} ^{\frac{n}{T}} \sum_{j = 1} ^{\frac{n}{T}} \gcd(a_{iT}, a_{jT}) \sum_{d \mid T} d \mu(\frac{T}{d})\\ \sum_{T = 1} ^{n} \phi(T) \sum_{i = 1} ^{\frac{n}{T}} \sum_{j = 1} ^{\frac{n}{T}} \gcd(a_{iT}, a_{jT})\\ i=1∑nj=1∑ngcd(ai,aj)gcd(i,j)d=1∑ndi=1∑dnj=1∑dngcd(aid,ajd)[gcd(i,j)=1]d=1∑ndk=1∑dnμ(k)i=1∑kdnj=1∑kdngcd(aikd,ajkd)T=kdT=1∑ni=1∑Tnj=1∑Tngcd(aiT,ajT)d∣T∑dμ(dT)T=1∑nϕ(T)i=1∑Tnj=1∑Tngcd(aiT,ajT)
我们考虑设f(n,T)=∑i=1nT∑j=1nTgcd(aiT,ajT)f(n, T) = \sum\limits_{i = 1} ^{\frac{n}{T}} \sum\limits_{j = 1} ^{\frac{n}{T}} \gcd(a_{iT}, a_{jT})f(n,T)=i=1∑Tnj=1∑Tngcd(aiT,ajT),g(x)g(x)g(x)为i∈[T,2T,…,nTT]i \in [T, 2T, \dots, \frac{n}{T} T]i∈[T,2T,…,TnT]时xxx的出现次数。
f(n,T)=∑i=1m∑j=1mg(i)g(j)gcd(i,j),(m=105)∑d=1md∑i=1md∑j=1mdg(id)g(jd)[gcd(i,j)=1]∑d=1md∑k=1ndμ(k)(∑i=1mkdg(ikd))2T=kd∑T=1mϕ(T)(∑i=1mTg(iT))2f(n, T) = \sum_{i = 1} ^{m} \sum_{j = 1} ^{m} g(i) g(j) \gcd(i, j), (m = 10 ^ 5)\\ \sum_{d = 1} ^{m} d \sum_{i = 1} ^{\frac{m}{d}} \sum_{j = 1} ^{\frac{m}{d}} g(id) g(jd) [\gcd(i, j) = 1]\\ \sum_{d = 1} ^{m} d \sum_{k = 1} ^{\frac{n}{d}} \mu(k) \left( \sum_{i = 1} ^{\frac{m}{kd}} g(ikd) \right) ^ 2\\ T = kd\\ \sum_{T = 1} ^{m} \phi(T) \left( \sum_{i = 1} ^{\frac{m}{T}} g(iT) \right) ^ 2\\ f(n,T)=i=1∑mj=1∑mg(i)g(j)gcd(i,j),(m=105)d=1∑mdi=1∑dmj=1∑dmg(id)g(jd)[gcd(i,j)=1]d=1∑mdk=1∑dnμ(k)⎝⎛i=1∑kdmg(ikd)⎠⎞2T=kdT=1∑mϕ(T)⎝⎛i=1∑Tmg(iT)⎠⎞2
考虑重新定义g(n)g(n)g(n)表示为是nnn的倍数的数字有多少个,则上式可以直接写成:
∑T=1mϕ(T)g(T)2\sum_{T = 1} ^{m} \phi(T) g(T) ^ 2\\ T=1∑mϕ(T)g(T)2
由此我们可以在O(nlog2n)O(n \log ^ 2n)O(nlog2n)的时间内完成这题。
#include <bits/stdc++.h>using namespace std;const int N = 1e5 + 10, mod = 1e9 + 7;int prime[N], phi[N], a[N], n, cnt;int sum[N], m;bool st[N];vector<int> fac[N];inline int add(int x, int y) {return x + y < mod ? x + y : x + y - mod;
}inline int sub(int x, int y) {return x >= y ? x - y : x - y + mod;
}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 = 1; 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++) {for (int j = i; j < N; j += i) {fac[j].push_back(i);}}
}int f(int n, int T) {int ans = 0;for (int i = T; i <= n; i += T) {for (auto it : fac[a[i]]) {ans = sub(ans, 1ll * phi[it] * sum[it] % mod * sum[it] % mod);sum[it]++;ans = add(ans, 1ll * phi[it] * sum[it] % mod * sum[it] % mod);}}for (int i = T; i <= n; i += T) {for (auto it : fac[a[i]]) {sum[it]--;}}return ans;
}int main() {// freopen("in.txt", "r", stdin);// freopen("out.txt", "w", stdout);init();scanf("%d", &n);for (int i = 1; i <= n; i++) {scanf("%d", &a[i]);}int ans = 0;for (int T = 1; T <= n; T++) {ans = add(ans, 1ll * phi[T] * f(n, T) % mod);}printf("%d\n", ans);return 0;
}