bzoj 2627: JZPKIL [伯努利数 Pollard-rho]

2627: JZPKIL

题意：求

\[\sum_{i=1}^n (n,i)^x [i,n]^y,\ [i,n] = lcm(i,n)
\]

\(n \le 10^{18},\ x,y\le 3000\)

---

本题带来了一种新技巧，n太大，转化成一个积性函数然后求这个积性函数，质因子分解利用积性，这养只与质因子的数量和指数有关

官方题解清橙上有

首先套路推♂倒

\[n^y \sum_{d \mid n} d^x \sum_{e \mid \frac{n}{d}} \mu(e) e^y \sum_{i=1}^{\frac{n}{de}} i^{y+1-k} \\
\]

这里使用\(D=de\)然后整除分块并不好做

代入伯努利数，交换求和顺序

\[\frac{n^y}{y+1} \sum_{k=0}^y \binom{y+1}{k} B_k^+ \sum_{d \mid n} d^x \sum_{e \mid \frac{n}{d}} \mu(e) e^y \sum_{i=1}^{\frac{n}{de}} i^{y+1-k} \\
\]

后面那块就是\((id^x *( (\mu \cdot id^y) * id^z)(n) = ( (\mu \cdot id^y) * id^x * id^z)(n)\)

考虑求这个积性函数

用Pollard-rho质因子分解n，然后求

\(( (\mu \cdot id^y) * id^x * id^z)(p^c)\)

这个很好求，\(\mu\)那里只能是\(1,p\)，剩下的直接\(O(c)\)计算就行了

质因子个数和指数都是log级的，所以复杂度大约是\(O(y^2 + n^{\frac{1}{4}}logn+ylog^2n)\)

#include <iostream>
#include <cstdio>
#include <cstring>
#include <algorithm>
#include <cmath>
using namespace std;
typedef long long ll;
const int N = 3005, P = 1e9+7, mo = P, inv2 = (P+1)/2;
inline int read(){
    char c=getchar(); int x=0,f=1;
    while(c<'0'||c>'9') {if(c=='-')f=-1;c=getchar();}
    while(c>='0'&&c<='9') {x=x*10+c-'0';c=getchar();}
    return x*f;
}

namespace qwq {
	ll mul(ll a, ll b, ll P) {
		ll t = (a * b - (ll) ((long double) a / P * b + 0.5) * P);
		return t<0 ? t+P : t;
	}
	ll Pow(ll a, ll b, ll P) {
		ll ans = 1; a %= P;
		for(; b; b>>=1, a = mul(a, a, P))
			if(b&1) ans = mul(ans, a, P);
		return ans;
	}
	bool witness(ll a, ll n, ll u, int t) {
		ll x = Pow(a, u, n), y = x;
		while(t--) {
			x = mul(x, x, n);
			if(x == 1 && y != 1 && y != n-1) return true;
			y = x;
		}
		return x != 1;
	}
	int p[10] = {2, 3, 5, 7, 11, 13, 17, 19, 23};
	bool m_r(ll n) {
		if(n == 2) return true;
		if(n <= 1 || ~n&1) return false;
		ll u = n-1, t = 0;
		while(~u&1) u >>= 1, t++;
		for(int i=0; i<9; i++) {
			if(p[i] == n) return true;
			if(witness(p[i], n, u, t)) return false;
		}
		return true;
	}
	ll gcd(ll a, ll b) {return !b ? a : gcd(b, a%b);}
	ll _rho(ll n, ll c) {
		int k = 2; ll x = rand()%n, y = x, d = 1;
		for(int i=1; d==1; i++) {
			x = (mul(x, x, n) + c) %n;
			d = gcd(n, x>y ? x-y : y-x);
			if(i == k) y = x, k <<= 1;
		}
		return d;
	}
	ll d[N];
	void p_r(ll n) {
		if(n == 1) return;
		if(m_r(n)) {d[++d[0]] = n; return;}
		ll t = n;
		while(t == n) t = _rho(n, rand() % (n-1) + 1);
		p_r(t); p_r(n/t);
	}
} using qwq::d;

ll Pow(ll a, ll b) {
	ll ans = 1; a %= P;
	for(; b; b>>=1, a=a*a%P)
		if(b&1) ans=ans*a%P;
	return ans;
}

int b[N];
ll inv[N], fac[N], facInv[N];
inline ll C(int n, int m) {return fac[n] * facInv[m] %P * facInv[n-m] %P;}
void init(int n) {
	inv[1] = fac[0] = facInv[0] = 1;
	for(int i=1; i<=n+1; i++) {
		if(i != 1) inv[i] = (P - P/i) * inv[P%i] %P;
		fac[i] = fac[i-1] * i %P;
		facInv[i] = facInv[i-1] * inv[i] %P;
	}

	b[0] = 1; b[1] = P-inv2;
	for(int m=2; m<=n; m++) if(!(m&1)) {
		for(int k=0; k<=m-1; k++) b[m] = (b[m] - C(m+1, k) * b[k]) %P;
		b[m] = b[m] * Pow(C(m+1, m), P-2) %P;
		if(b[m] < 0) b[m] += P;
	}
	b[1] = inv2;
}

struct meow{ll p; int k;} a[N]; int m;

ll p1[N], p2[N];
void solve(ll n, int x, int y) {
	d[0] = 0; qwq::p_r(n); 

	sort(d+1, d+d[0]+1);
	m = 0; a[++m] = (meow){d[1], 1};
	for(int i=2; i<=d[0]; i++) {
		if(d[i] == d[i-1]) a[m].k++;
		else a[++m] = (meow){d[i], 1};
	}
	//for(int i=1; i<=m; i++) printf("factor %d  %lld %d\n", i, a[i].p, a[i].k);

	ll ans = 0;
	for(int k=0; k<=y; k++) {
		ll t = C(y+1, k) * b[k] %mo;
		int z = y + 1 - k;
		for(int i=1; i<=m; i++) {
			ll p = a[i].p, t1 = 0, t2 = 0;
			int c = a[i].k;
			p1[0] = 1; ll u = p1[1] = Pow(p, x);
			p2[0] = 1; ll v = p2[1] = Pow(p, z);
			for(int j=2; j<=c; j++) p1[j] = p1[j-1] * u %mo, p2[j] = p2[j-1] * v %mo;
			for(int j=0; j<=c; j++) t1 = (t1 + p1[j] * p2[c-j]) %mo;
			for(int j=0; j<c; j++)  t2 = (t2 + p1[j] * p2[c-1-j]) %mo;
			t2 = (mo - Pow(p, y)) * t2 %mo;
			t = t * (t1 + t2) %mo;
		}
		ans = (ans + t) %mo;
	}
	ans = ans * Pow(n, y) %mo * Pow(y+1, mo-2) %mo;
	printf("%lld\n", ans);
}

int x, y; ll n;

int main() {
	freopen("in", "r", stdin);
	init(3001);
	int T = read();
	while(T--) {
		scanf("%lld", &n); x = read(); y = read();
		solve(n, x, y);
	}
}