所求的式子带除法,模意义下除法计算复杂度带 \(\log\) 太慢了,先改写成乘法:\(\sum_{i=1}^n i!\times i^{-k}\)。想求这个式子,最简单的思路就是对于每个整数 \(i\in[1,n]\),分别预处理出 \(i!\) 和 \(i^{-k}\) 的值,最后乘起来再 \(O(n)\) 暴力加起来就好了!
对于 \(i!\),注意到:
\[i!= \begin{cases} 1,&i=1\\ (i-1)!\times i,&i\ge 2\\ \end{cases} \]可以直接 \(O(n)\) 递推。
对于 \(i^{-k}\),注意到:
\[(ij)^{-k}=i^{-k}j^{-k} \]因此 \(i^{-k}\) 是关于 \(i\) 的完全积性函数,且该函数进行单点求值的复杂度为 \(O(\log n)\)。\(n\) 以内质数个数是 \(O(\frac{n}{\log n})\) 的,暴力求质数处的点值之后,可以使用欧拉筛 \(O(n)\) 求出。
总复杂度 \(O(n)\)。
// Problem: P11253 [GDKOI2023 普及组] 小学生数学题
// Contest: Luogu
// URL: https://www.luogu.com.cn/problem/P11253
// Memory Limit: 512 MB
// Time Limit: 1000 ms
//
// Powered by CP Editor (https://cpeditor.org)
//By: OIer rui_er
#include <bits/stdc++.h>
#define rep(x, y, z) for(int x = (y); x <= (z); ++x)
#define per(x, y, z) for(int x = (y); x >= (z); --x)
#define debug(format...) fprintf(stderr, format)
#define fileIO(s) do {freopen(s".in", "r", stdin); freopen(s".out", "w", stdout);} while(false)
#define endl '\n'
using namespace std;
typedef long long ll;
mt19937 rnd(std::chrono::duration_cast<std::chrono::nanoseconds>(std::chrono::system_clock::now().time_since_epoch()).count());
int randint(int L, int R) {
uniform_int_distribution<int> dist(L, R);
return dist(rnd);
}
template<typename T> void chkmin(T& x, T y) {if(y < x) x = y;}
template<typename T> void chkmax(T& x, T y) {if(x < y) x = y;}
template<int mod>
inline unsigned int down(unsigned int x) {
return x >= mod ? x - mod : x;
}
template<int mod>
struct Modint {
unsigned int x;
Modint() = default;
Modint(unsigned int x) : x(x) {}
friend istream& operator>>(istream& in, Modint& a) {return in >> a.x;}
friend ostream& operator<<(ostream& out, Modint a) {return out << a.x;}
friend Modint operator+(Modint a, Modint b) {return down<mod>(a.x + b.x);}
friend Modint operator-(Modint a, Modint b) {return down<mod>(a.x - b.x + mod);}
friend Modint operator*(Modint a, Modint b) {return 1ULL * a.x * b.x % mod;}
friend Modint operator/(Modint a, Modint b) {return a * ~b;}
friend Modint operator^(Modint a, int b) {Modint ans = 1; for(; b; b >>= 1, a *= a) if(b & 1) ans *= a; return ans;}
friend Modint operator~(Modint a) {return a ^ (mod - 2);}
friend Modint operator-(Modint a) {return down<mod>(mod - a.x);}
friend Modint& operator+=(Modint& a, Modint b) {return a = a + b;}
friend Modint& operator-=(Modint& a, Modint b) {return a = a - b;}
friend Modint& operator*=(Modint& a, Modint b) {return a = a * b;}
friend Modint& operator/=(Modint& a, Modint b) {return a = a / b;}
friend Modint& operator^=(Modint& a, int b) {return a = a ^ b;}
friend Modint& operator++(Modint& a) {return a += 1;}
friend Modint operator++(Modint& a, int) {Modint x = a; a += 1; return x;}
friend Modint& operator--(Modint& a) {return a -= 1;}
friend Modint operator--(Modint& a, int) {Modint x = a; a -= 1; return x;}
friend bool operator==(Modint a, Modint b) {return a.x == b.x;}
friend bool operator!=(Modint a, Modint b) {return !(a == b);}
};
const int N = 2e7 + 5, mod = 998244353;
typedef Modint<mod> mint;
int n, k, tab[N], p[N], pcnt;
mint fac[N], invpw[N], ans;
int main() {
ios::sync_with_stdio(false);
cin.tie(0); cout.tie(0);
cin >> n >> k;
fac[0] = fac[1] = 1;
rep(i, 2, n) fac[i] = fac[i - 1] * i;
tab[1] = 1;
invpw[1] = ~(mint(1) ^ k);
rep(i, 2, n) {
if(!tab[i]) {
p[++pcnt] = i;
invpw[i] = ~(mint(i) ^ k);
}
for(int j = 1; j <= pcnt && i * p[j] <= n; ++j) {
tab[i * p[j]] = 1;
invpw[i * p[j]] = invpw[i] * invpw[p[j]];
if(i % p[j] == 0) break;
}
}
rep(i, 1, n) ans += fac[i] * invpw[i];
cout << ans << endl;
return 0;
}
标签:return,GDKOI2023,int,题解,friend,operator,Modint,P11253,mod
From: https://www.cnblogs.com/ruierqwq/p/18533851/LG-P11253