Lady Layton with Math（杜教筛）

Lady Layton with Math

∑ i = 1 n ∑ j = 1 n ϕ ( g c d ( i , j ) ) ∑ d = 1 n ϕ ( d ) ∑ i = 1 n ∑ j = 1 n [ g c d ( i , j ) = d ] ∑ d = 1 n ϕ ( d ) ∑ i = 1 n d ∑ j = 1 n d [ g c d ( i , j ) = 1 ] ∑ d = 1 n ϕ ( d ) ( 2 ∑ i = 1 n d ∑ j = 1 i [ g c d ( i , j ) = 1 ] − 1 ) ∑ d = 1 n ϕ ( d ) ( 2 ∑ i = 1 n d ϕ ( i ) − 1 ) \sum_{i = 1} ^{n} \sum_{j = 1} ^{n} \phi(gcd(i, j))\\ \sum_{d = 1} ^{n} \phi(d) \sum_{i = 1} ^{n} \sum_{j = 1} ^{n} [gcd(i, j) = d]\\ \sum_{d = 1} ^{n} \phi(d) \sum_{i = 1} ^{\frac{n}{d}} \sum_{j = 1} ^{\frac{n}{d}} [gcd(i, j) = 1]\\ \sum_{d = 1} ^{n} \phi(d) \left(2 \sum_{i = 1} ^{\frac{n}{d}} \sum_{j = 1} ^{i} [gcd(i, j) = 1] - 1 \right)\\ \sum_{d = 1} ^{n} \phi(d)\left(2 \sum_{i = 1} ^{\frac{n}{d}} \phi(i) - 1 \right)\\ i=1∑n​j=1∑n​ϕ(gcd(i,j))d=1∑n​ϕ(d)i=1∑n​j=1∑n​[gcd(i,j)=d]d=1∑n​ϕ(d)i=1∑dn​​j=1∑dn​​[gcd(i,j)=1]d=1∑n​ϕ(d)⎝⎛​2i=1∑dn​​j=1∑i​[gcd(i,j)=1]−1⎠⎞​d=1∑n​ϕ(d)⎝⎛​2i=1∑dn​​ϕ(i)−1⎠⎞​

/*
  Author : lifehappy
*/
#include <bits/stdc++.h>

using namespace std;

typedef long long ll;

const int N = 2e6 + 10, mod = 1e9 + 7;

int prime[N], cnt, n;

ll phi[N];

bool st[N];

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++) {
        phi[i] = (phi[i] + phi[i - 1]) % mod;
    }
}

unordered_map<int, ll> ans_s;

ll Djs(int n) {
    if(n < N) return phi[n];
    if(ans_s.count(n)) return ans_s[n];
    ll ans = 1ll * n * (n + 1) / 2 % mod;
    for(int l = 2, r; l <= n; l = r + 1) {
        r = n / (n / l);
        ans = (ans - (r - l + 1) * Djs(n / l) % mod + mod) % mod;
    }
    return ans_s[n] = ans;
}

int main() {
    // freopen("in.txt", "r", stdin);
    // freopen("out.txt", "w", stdout);
    // ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);
    int T;
    init();
    scanf("%d", &T);
    while(T--) {
        ll ans = 0;
        scanf("%d", &n);
        for(int l = 1, r; l <= n; l = r + 1) {
            r = n / (n / l);
            ans = (ans + 1ll * (Djs(r) - Djs(l - 1)) * (2ll * Djs(n / l) - 1) % mod + mod) % mod;
        }
        printf("%lld\n", ans);
    }
    return 0;
}