HDU 6706 huntian oy （欧拉函数 + 杜教筛）

huntian oy

推式子

$$\begin{gathered} \sum _{i=1}^n\sum _{j=1}^{i}gcd(i^a-j^a,i^b-j^b)(gcd(i,j)==1) \\ =\sum _{i=1}^n\sum _{j=1}^i(i-j)(gcd(i,j)==1) \\ =\sum _{i=1}^{n}i\sum _{j=1}^i(gcd(i,j)==1)-\sum _{i=1}^n\sum _{j=1}^{i}j(gcd(i,j)==1) \\ =\sum _{i=1}^{n}i\varphi (i)-\sum _{i=1}^n\frac{i\varphi (i)+(i==1)}{2} \\ =\sum _{i=1}^n\frac{i\varphi (i)-(i==1)}{2} \\ 然后套路地变成求S(n)=\sum _{i=1}^{n}i\varphi (i) \\ g(n)=n\varphi (n) \\ 这里直接套用杜教筛化简得到g(1)S(n)=\sum _{i=1}^n(f\ast g)(i)-\sum _{i=2}f(i)S(\frac{n}{i}) \\ (f\ast g)(i)=\sum _{d\mid n}f(d)\ast g(\frac{n}{d})=\sum _{d\mid n}f(d)\ast \frac{n}{d}\varphi (\frac{n}{d}) \\ 另f(d)=d，则有S(n)=\sum _{i=1}^n{i}^2-\sum _{i=2}iS(\frac{n}{i}) \\ 然后直接套杜教筛即可。 \end{gathered}$$

代码

/*
  Author : lifehappy
*/
#pragma GCC optimize(2)
#pragma GCC optimize(3)
#include 

#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 | 1

using 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 = 1e6 + 10, mod = 1e9 + 7, inv2 = 500000004, inv6 = 166666668;

int prime[N], cnt;

ll phi[N];

bool st[N];

ll quick_pow(ll a, ll n) {
    ll ans = 1;
    while(n) {
        if(n & 1) ans = ans * a % mod;
        n >>= 1;
        a = a * a % mod;
    }
    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 && 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] = (1ll * phi[i] * i + phi[i - 1]) % mod;
    }
}

ll calc(int n) {
    return 1ll * n * (n + 1) % mod * (2 * n + 1) % mod * inv6 % mod;
}

unordered_map<int, ll> ans_s;

ll S(int n) {
    if(n < N) return phi[n];
    if(ans_s.count(n)) return ans_s[n];
    ll ans = calc(n);
    for(int l = 2, r; l <= n; l = r + 1) {
        r = n / (n / l);
        ans = (ans - 1ll * (r + l) * (r - l + 1) / 2 % mod * S(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);
    init();
    int T = read();
    while(T--) {
        int n = read(), a = read(), b = read();
        printf("%lld\n", (S(n) - 1 + mod) % mod * inv2 % mod);
    }
	return 0;
}