莫比乌斯反演例题集---（自用）

P3455 [POI2007]ZAP-Queries

$$求解\sum _{i=1}^n\sum _{j=1}^m[gcd(i,j)=k]$$
$反演过程：$
$$\sum _{i=1}^n\sum _{j=1}^m[gcd(i,j)=k]$$

$$\sum _{i=1}^{\lfloor \frac{n}{k}\rfloor }\sum _{j=1}^{\lfloor \frac{m}{k}\rfloor }\epsilon [gcd(i,j)=1]$$

$枚举d：$

$$\sum _{d=1}^{\lfloor min(\lfloor \frac{n}{k}\rfloor ,\lfloor \frac{m}{k}\rfloor )\rfloor }\sum _{d|T}\mu (d)\lfloor \frac{n}{T}\rfloor \lfloor \frac{m}{T}\rfloor$$

$T换kd得：$

$$\sum _{d=1}^{\lfloor min(\lfloor \frac{n}{k}\rfloor ,\lfloor \frac{m}{k}\rfloor )\rfloor }\mu (d)\lfloor \frac{n}{kd}\rfloor \lfloor \frac{m}{kd}\rfloor$$

$先预处理莫比乌斯函数，然后整数分块做，复杂度为O(n+T\sqrt{n})$

Code

#include 

using namespace std;

typedef long long ll;
typedef long double ld;
typedef pair<int, int> pdd;

#define INF 0x7f7f7f
#define mem(a, b) memset(a , b , sizeof(a))
#define FOR(i, x, n) for(int i = x;i <= n; i++)

// const ll mod = 998244353;
// const int maxn = 1e5 + 10;
// const double eps = 1e-6;

const int N = 5e5 + 10;

int mu[N]; // 莫比乌斯函数
bool is_prime[N];
int prime[N];
int cnt;

ll sum[N]; // 记录莫比乌斯函数前缀和


void Mobi() // 莫比乌斯函数初始化
{
    mu[1] = 1;
    is_prime[0] = is_prime[1] = true;
    for(int i = 2;i < N; i++) {
        if (!is_prime[i]) {
            mu[i] = -1;
            prime[++cnt] = i;
        }
        for (int j = 1; j <= cnt && i * prime[j] < N; j++) {
            is_prime[i * prime[j]] = true;
            if (i % prime[j] == 0) {
                mu[i * prime[j]] = 0;
                break;
            }
            mu[i * prime[j]] = -mu[i];
        }
    }

    for(int i = 1;i < N; i++) {
        sum[i] = sum[i - 1] + mu[i];
    }
}

ll k;

ll Ans(ll n, ll m) {
    ll ans = 0;
    n /= k, m /= k;
    int mx = min(n, m);
    for(int l = 1, r;l <= mx; l = r + 1) {
        r = min(n / (n / l), m / (m / l));
        ans += (n / l) * (m / l) * (sum[r] - sum[l - 1]);
    }
    return ans;
}

void solve() {
    Mobi();
    int T;
    cin >> T;
    while(T--) {
        ll n, m;
        cin >> n >> m >> k;
        cout << Ans(n, m) << endl;
    }
}


signed main() {
    ios_base::sync_with_stdio(false);
    //cin.tie(nullptr);
    //cout.tie(nullptr);
#ifdef FZT_ACM_LOCAL
    freopen("in.txt", "r", stdin);
    freopen("out.txt", "w", stdout);
    signed test_index_for_debug = 1;
    char acm_local_for_debug = 0;
    do {
        if (acm_local_for_debug == '$') exit(0);
        if (test_index_for_debug > 20)
            throw runtime_error("Check the stdin!!!");
        auto start_clock_for_debug = clock();
        solve();
        auto end_clock_for_debug = clock();
        cout << "Test " << test_index_for_debug << " successful" << endl;
        cerr << "Test " << test_index_for_debug++ << " Run Time: "
             << double(end_clock_for_debug - start_clock_for_debug) / CLOCKS_PER_SEC << "s" << endl;
        cout << "--------------------------------------------------" << endl;
    } while (cin >> acm_local_for_debug && cin.putback(acm_local_for_debug));
#else
    solve();
#endif
    return 0;
}

P2257 YY的GCD