1238 最小公倍数之和 V3

推式子

$$\begin{gathered} \sum _{i=1}^n\sum _{j=1}^{n}lcm(i,j) \\ =\sum _{i=1}^n\sum _{j=1}^n\frac{ij}{gcd(i,j)} \\ =\sum _{d=1}^n\sum _{i=1}^n\sum _{j=1}^n\frac{ij}{d}(gcd(i,j)==d) \\ =\sum _{d=1}^{n}d\sum _{i=1}^{\frac{n}{d}}\sum _{j=1}^{\frac{n}{d}}ij(gcd(i,j)==1) \\ =\sum _{d=1}^{n}d\sum _{i=1}^{\frac{n}{d}}\sum _{j=1}^{\frac{n}{d}}ij\sum _{k\mid gcd(i,j)}\mu (k) \\ =\sum _{d=1}^{n}d\sum _{k=1}^{\frac{n}{d}}{k}^2\mu (k)\sum _{i=1}^{\frac{n}{dk}}\sum _{j=1}^{\frac{n}{dk}}ij \end{gathered}$$

化简到这里好像能通过两次数论分块得到我们的答案，但是复杂度是$O(n)$的，显然不行，所以我们考虑另一种化简方法。
$$\begin{gathered} \sum _{i=1}^n\sum _{j=1}^{n}lcm(i,j) \\ =\sum _{i=1}^n\sum _{j=1}^n\frac{ij}{gcd(i,j)} \\ =\sum _{d=1}^n\sum _{i=1}^n\sum _{j=1}^n\frac{ij}{d}(gcd(i,j)==d) \\ =\sum _{d=1}^{n}d\sum _{i=1}^{\frac{n}{d}}\sum _{j=1}^{\frac{n}{d}}ij(gcd(i,j)==1) \\ 这里两个的上届都是\frac{n}{d},所以我们可以分类讨论一下，i>j,i==j,i<j，由于i>j,i<j可以合并，所以上式变成 \\ =\sum _{d=1}^{n}d(2\sum _{i=1}^{\frac{n}{d}}i(\sum _{j=1}^{i}j\times (gcd(i,j)==1))-1) \\ 根据欧拉函数定理，我们可以再次化简 \\ =\sum _{d=1}^{n}d(2\sum _{i=1}^{\frac{n}{d}}i(\frac{i\varphi (i)+(i==1)}{2})-1) \\ =\sum _{d=1}^{n}d\sum _{i=1}^{\frac{n}{d}}{i}^2\varphi (i) \\ 接下来就是按照套路用杜教筛求解\sum _{i=1}^{\frac{n}{d}}{i}^2\varphi (i)了 \\ S(n)=\sum _{i=1}^n{i}^2\varphi (i)=\sum _{i=1}^{n}f(i) \\ \sum _{i=1}^n(f\ast g)(i) \\ =\sum _{i=1}^{n}g(i)S(\frac{n}{i}) \\ g(1)S(n)=\sum _{i=1}^n(f\ast g)(i)-\sum _{d=2}^{n}g(i)S(\frac{n}{d}) \\ (f\ast g)(i)=\sum _{d\mid i}f(d)g(\frac{i}{d})=\sum _{d\mid i}{d}^2\varphi (d)g(\frac{i}{d}) \\ 容易想到\sum _{d\mid i}\varphi (d)=i,所以如果可以提出d^2那就完美了，我们可以另g(i)=i^2，上式变成 \\ \sum _{d\mid i}{d}^2\varphi (d)\frac{i^2}{d^2}=i^2\sum _{d\mid i}\varphi (d)=i^3 \\ S(n)=\sum _{i=1}^n{i}^3-\sum _{d=1}^n{d}^{2}S(\frac{n}{d}) \\ =\frac{n^2(n+1{)}^2}{4}-\sum _{d=1}^n{d}^{2}S(\frac{n}{d}) \end{gathered}$$
最终化简的式子${\sum }_{d=1}^{n}dS(\frac{n}{d})$

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 = 8e6 + 10, mod = 1000000007;ll phi[N], n, inv4, inv6, 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] + 1ll * i * i % mod * phi[i] % mod) % mod;}inv2 = quick_pow(2, mod - 2, mod), inv4 = quick_pow(4, mod - 2, mod), inv6 = quick_pow(6, mod - 2, mod);
}ll calc1(ll x) {x %= mod;return 1ll * x * x % mod * (x + 1) % mod * (x + 1) % mod * inv4 % mod;
}ll calc2(ll x) {x %= mod;return x * (x + 1) % mod * (2ll * x + 1) % mod * inv6 % mod;
}unordered_map<ll, 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 = calc1(x);for(ll l = 2, r; l <= x; l = r + 1) {r = x / (x / l);ans -= (calc2(r) - calc2(l - 1)) * get_phi(x / l) % mod;ans = (ans % mod + mod) % mod;}return ans_phi[x] = ans;
}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();init();ll ans = 0;for(ll l = 1, r; l <= n; l = r + 1) {r = n / (n / l);ans += (l + r) % mod * (r - l + 1) % mod * inv2 % mod * get_phi(n / l) % mod;ans %= mod; }printf("%lld\n", ans);return 0;
}