Product

$$\begin{gathered} \sum _{i=1}^n\sum _{j=1}^n\sum _{k=1}^n\gcd (i,j)[k\mid \gcd (i,j)] \\ \sum _{k=1}^n\sum _{i=1}^{\frac{n}{k}}\sum _{j=1}^{\frac{n}{k}}\gcd (ik,jk) \\ \sum _{k=1}^{n}k\sum _{i=1}^{\frac{n}{k}}\sum _{j=1}^{\frac{n}{k}}\gcd (i,j) \\ \sum _{k=1}^{n}k\sum _{d=1}^{\frac{n}{k}}d\sum _{i=1}^{\frac{n}{kd}}\sum _{j=1}^{\frac{n}{kd}}[\gcd (i,j)=1] \\ \sum _{k=1}^{n}k\sum _{d=1}^{\frac{n}{k}}d?\sum _{i=1}^{\frac{n}{kd}}2?(i)?1? \\ T=kd \\ \sum _{T=1}^n\sum _{k\mid T}k\times \frac{T}{k}?\sum _{i=1}^{\frac{n}{T}}2?(i)?1? \\ \sum _{T=1}^{n}T{\sigma }_0(T)?\sum _{i=1}^{\frac{n}{T}}2?(i)?1? \end{gathered}$$

設$f(n)=\sum _{i=1}^{n}i{\sigma }_0(i)$。
$$\begin{gathered} \sum _{i=1}^{n}i\sum _{d\mid i} \\ \sum _{d=1}^n\sum _{i=1}^{\frac{n}{d}}i\times d \\ \sum _{d=1}^{n}d\frac{(\frac{n}{d}+1)\frac{n}{d}}{2} \end{gathered}$$

#include <bits/stdc++.h>using namespace std;const int N = 1e6 + 10;int f[N], num[N], phi[N], prime[N], primes[N], cnt;bool st[N];int n, m, p, mod;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() {f[1] = phi[1] = 1;for (int i = 2; i < N; i++) {if (!st[i]) {prime[++cnt] = i;phi[i] = i - 1;f[i] = 2;num[i] = 1;primes[i] = i;}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];f[i * prime[j]] = f[i / primes[i]] * (num[i] + 2);num[i * prime[j]] = num[i] + 1;primes[i * prime[j]] = primes[i] * prime[j];break;}phi[i * prime[j]] = phi[i] * (prime[j] - 1);f[i * prime[j]] = f[i] * 2;num[i * prime[j]] = 1;primes[i * prime[j]] = prime[j];}}for (int i = 1; i < N; i++) {phi[i] = add(phi[i], phi[i - 1]);f[i] = add(1ll * i * f[i] % mod, f[i - 1]);} }unordered_map<int, int> Phi;int Djs_phi(int n) {if (n < N) {return phi[n];}if (Phi.count(n)) {return Phi[n];}int ans = 1ll * n * (n + 1) / 2 % mod;for (int l = 2, r; l <= n; l = r + 1) {r = n / (n / l);ans = sub(ans, Djs_phi(n / l) * (r - l + 1) % mod);}return Phi[n] = ans; }unordered_map<int, int> f1;int calc1(int l, int r) {return 1ll * (l + r) * (r - l + 1) / 2 % mod; }int F(int n) {if (n < N) {return f[n];}if (f1.count(n)) {return f1[n];}int ans = 0;for (int l = 1, r; l <= n; l = r + 1) {r = n / (n / l);ans = add(ans, 1ll * (n / l) * (n / l + 1) / 2 % mod * calc1(l, r) % mod);}return f1[n] = ans; }int calc(int n) {return sub(2ll * Djs_phi(n) % mod, 1); }int quick_pow(int a, int n) {int ans = 1;while (n) {if (n & 1) {ans = 1ll * ans * a % p;}n >>= 1;a = 1ll * a * a % p;}return ans; }int main() {// freopen("in.txt", "r", stdin);// freopen("ou.txt", "w", stdout);scanf("%d %d %d", &n, &m, &p);mod = p - 1;init();int ans = 0;for (int l = 1, r; l <= n; l = r + 1) {r = n / (n / l);ans = add(ans, 1ll * sub(F(r), F(l - 1)) * calc(n / l) % mod);}printf("%d\n", quick_pow(m, ans));return 0; }

總結

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