[CodeForces 439E] Devu and Birthday Celebration（莫比乌斯反演） | 错题本

原创于 2020-09-04 12:33:29 发布 · 235 阅读

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#莫比乌斯函数

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本文详细解析了CodeForces439E题目“Devu and Birthday Celebration”的解决方案，介绍了莫比乌斯函数的重要性质及其在解决此题中的应用。通过预处理莫比乌斯函数并结合隔板法，实现了一个高效的算法。

文章目录

题目
分析
代码

题目

[CodeForces 439E] Devu and Birthday Celebration

分析

莫比乌斯函数比较重要的性质： $\mu * 1 = \varepsilon$ 即 $\sum_{d | n}\mu(d) = [n = 1]$

证明：
设 $\prod_{i = 1}^{m} p_i^{\alpha_i}$ ， $\prod_{i = 1}^mp_i$ ，有 $\sum_{d | n}\mu(d) = \sum_{d | n'}\mu(d) = \sum_{i = 0}^{m} \binom{m}{i} (-1)^i$ 由二项式定理，当 $n = 1$ 时，上式为 $1$ ；当 $\neq 1$ 时，上式为 $1 + (-1)]^m = 0$ 。

$\begin{aligned} &\sum_{a_1 = 1}^n \sum_{a_2 = 1}^n \cdots \sum_{a_f = 1}^n \left[\sum_{i = 1}^fa_i = n\right] \left[\gcd_{i = 1}^f a_i = 1\right] \\ =& \sum_{a_1 = 1}^n \sum_{a_2 = 1}^n \cdots \sum_{a_f = 1}^n \left[\sum_{i = 1}^f a_i = n\right] \sum_{d |\text{gcd}_{i = 1}^f a_i} \mu(d) \\ =& \sum_{d | n} \mu(d) \sum_{k_1 =1}^{\frac{n}{d}} \sum_{k_2 =1}^{\frac{n}{d}} \cdots \sum_{k_f =1}^{\frac{n}{d}} \left[\sum_{i = 1}^f k_i = \frac{n}{d} \right]\end{aligned}$ （转换枚举方式， $a_i = d\cdot k_i$ ，由于 $d | a_i$ ，所以 $\sum_{i = 1}^f a_i$ 即 $d ∣ n$ ）

预处理莫比乌斯函数，然后 $O(\sqrt{n})$ 枚举因数 + 隔板法即可。

代码

#include <bits/stdc++.h>

typedef long long LL;

template <const int _MOD> struct ModNumber {
    int x;
    inline ModNumber() { x = 0; }
    inline ModNumber(const int &y) { x = y; }
    inline int Int() { return x; }
    inline ModNumber Pow(int y) const { register int ret = 1, tmp = x; while (y) { if (y & 1) ret = ((LL)ret * tmp) % _MOD; y >>= 1; tmp = ((LL)tmp * tmp) % _MOD; } return ModNumber(ret); }
    inline bool operator == (const ModNumber &y) const { return x == y.x; }
    inline bool operator != (const ModNumber &y) const { return x != y.x; }
    inline bool operator < (const ModNumber &y) const { return x < y.x; }
    inline bool operator > (const ModNumber &y) const { return x > y.x; }
    inline bool operator <= (const ModNumber &y) const { return x <= y.x; }
    inline bool operator >= (const ModNumber &y) const { return x >= y.x; }
    inline ModNumber operator - () const { return _MOD - x; }
    inline ModNumber operator + (const ModNumber &y) const { return (x + y.x >= _MOD) ? (x + y.x - _MOD) : (x + y.x); }
    inline ModNumber operator - (const ModNumber &y) const { return (x - y.x < 0) ? (x - y.x + _MOD) : (x - y.x); }
    inline ModNumber operator * (const ModNumber &y) const { return ModNumber((LL)x * y.x % _MOD); }
    inline ModNumber operator / (const ModNumber &y) const { return *this * y.Pow(_MOD - 2); }
    inline ModNumber operator ^ (const int &y) const { return Pow(y); }
    inline void operator += (const ModNumber &y) { *this = *this + y; }
    inline void operator *= (const ModNumber &y) { *this = *this * y; }
    inline void operator -= (const ModNumber &y) { *this = *this - y; }
    inline void operator /= (const ModNumber &y) { *this = *this / y; }
    inline void operator ^= (const int &y) const { *this = *this ^ y; }
    inline bool operator == (const int &y) const { return x == y; }
    inline bool operator != (const int &y) const { return x != y; }
    inline bool operator < (const int &y) const { return x < y; }
    inline bool operator > (const int &y) const { return x > y; }
    inline bool operator <= (const int &y) const { return x <= y; }
    inline bool operator >= (const int &y) const { return x >= y; }
    inline ModNumber operator + (const int &y) const { return (x + y >= _MOD) ? (x + y - _MOD) : (x + y); }
    inline ModNumber operator - (const int &y) const { return (x - y < 0) ? (x - y + _MOD) : (x - y); }
    inline ModNumber operator * (const int &y) const { return ModNumber((LL)x * y % _MOD); }
    inline ModNumber operator / (const int &y) const { return *this * ModNumber(y).Pow(_MOD - 2); }
    inline void operator += (const int &y) { *this = *this + y; }
    inline void operator *= (const int &y) { *this = *this * y; }
    inline void operator -= (const int &y) { *this = *this - y; }
    inline void operator /= (const int &y) { *this = *this / y; }
};

const int MAXN = 2 * 100000;
const int MOD = 1000000007;

typedef ModNumber<MOD> Int;

Int Fac[MAXN + 5], Inv[MAXN + 5];

Int Mu[MAXN + 5];
bool Vis[MAXN + 5];
std::vector<int> Primes;

void Init(int n) {
	Mu[1] = 1;
	for (int i = 2; i <= n; i++) {
		if (!Vis[i])
			Mu[i] = MOD - 1, Primes.push_back(i);
		for (int j = 0; j < (int)Primes.size() && i * Primes[j] <= n; j++) {
			Vis[i * Primes[j]] = true;
			if (i % Primes[j] == 0) {
				Mu[i * Primes[j]] = 0;
				break;
			}
			Mu[i * Primes[j]] = -Mu[i];
		}
	}
	Fac[0] = 1;
	for (int i = 1; i <= n; i++)
		Fac[i] = Fac[i - 1] * i;
	Inv[n] = Fac[n] ^ (MOD - 2);
	for (int i = n - 1; i >= 0; i--)
		Inv[i] = Inv[i + 1] * (i + 1);
}

Int C(int n, int m) {
	if (n < m) return 0;
	return Fac[n] * Inv[m] * Inv[n - m];
}

int main() {
	Init(MAXN);
	int Q; scanf("%d", &Q);
	while (Q--) {
		Int Ans = 0;
		int N, F; scanf("%d%d", &N, &F);
		for (int i = 1; i * i <= N; i++) {
			if (N % i) continue;
			int j = N / i; Ans += Mu[i] * C(j - 1, F - 1);
			if (j != i) Ans += Mu[j] * C(i - 1, F - 1);
		}
		printf("%d\n", Ans.x);
	}
	return 0;
}