LG P4449 & JZOJ 于神之怒
\(\text{Problem}\)
JZOJ上,求
\[\sum_{i=1}^n \sum_{j=1}^m \gcd(i,j)^k
\]
对 \(10^9+7\) 取模
\(n,m,k \le 5 \times 10^6\)
LG 上,是一个加强版,有 \(T(T\le 2 \times 10^3)\) 组数据
\(\text{Analysis}\)
依照套路的方法,我们可以推出
\[Ans = \sum_{p=1} p^k \sum_{d=1} \mu(d) \lfloor \frac{n}{pd} \rfloor \lfloor \frac{m}{pd} \rfloor
\]
若只有一组数据,那么
数论分块套数论分块 \(O(n^{\frac{3}{4}})\) 即可
加上线筛 \(O(n)\)
\(\text{Code}\)
#include<cstdio>
#include<iostream>
#define LL long long
#define re register
using namespace std;
const int N = 5e6, P = 1e9 + 7;
int n, m, k, totp, pr[N], vis[N + 5], sum[N + 5], pk[N + 5];
inline int fpow(LL x, LL y)
{
LL res = 1;
for(; y; y >>= 1)
{
if (y & 1) res = res * x % P;
x = x * x % P;
}
return res;
}
inline void Euler()
{
vis[1] = sum[1] = pk[1] = 1;
for(re int i = 2; i <= N; i++)
{
if (!vis[i]) pr[++totp] = i, sum[i] = -1, pk[i] = fpow(i, k);
for(re int j = 1; j <= totp && i * pr[j] <= N; j++)
{
vis[i * pr[j]] = 1, pk[i * pr[j]] = (LL)pk[i] * pk[pr[j]] % P;
if (!(i % pr[j])) break;
sum[i * pr[j]] = -sum[i];
}
}
for(re int i = 1; i <= N; i++) sum[i] += sum[i - 1], pk[i] = (pk[i] + pk[i - 1]) % P;
}
inline int F(int n, int m)
{
LL res = 0;
for(re int l = 1, r; l <= min(n, m); l = r + 1)
{
r = min(n / (n / l), m / (m / l));
res = (res + (LL)(sum[r] - sum[l - 1] + P) * (n / l) % P * (m / l)) % P;
}
return res;
}
int main()
{
scanf("%d%d%d", &n, &m, &k);
Euler();
LL ans = 0;
for(re int l = 1, r; l <= min(n, m); l = r + 1)
{
r = min(n / (n / l), m / (m / l));
ans = (ans + (LL)(pk[r] - pk[l - 1] + P) * F(n / l, m / l) % P) % P;
}
printf("%lld\n", ans);
}
但LG上有多组数据,显然太慢了
同样套路地 \(T=pd\)
然后这个式子成了
\[\sum_{T=1} \lfloor \frac{n}{T} \rfloor \lfloor \frac{m}{T} \rfloor \sum_{d|T} d^k \mu(\frac{T}{d})
\]
\(g(d)=d^k\) 显然是个积性函数,然后 \(G=g * mu\) 也是个积性函数
于是我们考虑线筛预处理 \(G\),然后数论分快做到单次 \(O(\sqrt n)\)
根据积性函数性质有 \(G(d) = \prod_{i=1} G({p_i}^{c_i})\)
然后我们思考什么样的数有贡献
\[G(n) = \prod_{i=1} \sum_{j=0}^{c_i} \mu({p_i}^{j}) {p_i}^{(c_i-j)k}
\]
因为 \(\mu\) 的性质,我们知道,只有当 \(j=0\) 或 \(j=1\) 时有贡献,于是有
\[\begin{aligned}
G(n)
&= \prod_{i=1} \mu(1) {p_i}^{c_i k} + \mu(p_i) {p_i}^{(c_i-1)k} \\
&= \prod_{i=1} {p_i}^{c_i k} - {p_i}^{(c_i-1)k} \\
&= \prod_{i=1} {p_i}^{(c_i-1) k}({p_i}^k-1)
\end{aligned}
\]
当 \(c_i = 1\) 的时候,就是质数的时候,\(G(p)=p^k-1\)
因为 \(G\) 是积性函数,所以 \(G(ab)=G(a)G(b)(\gcd(a,b)=1)\)
若 \(a,b\) 不互质,因为在线筛时枚举质数,所以 \(b\in \mathbb P\),设 \(a = a' p^c(\gcd(a,a')=1)\)
那么 \(G(ab)=G(a')G(p^{c+1})=G(a')p^{ck}(p^k-1)\)
线筛过程中 \(p^{(c-1)k}(p^k-1)\) 已计入 \(G(ab)\) 中,所以本次再乘上 \(p^k\) 即可
综上
\[G(ab)=
\begin{cases}
G(a)G(b) & \gcd(a,b)=1 \\
G(a)b^k & \gcd(a,b)>1
\end{cases}
\]
线筛即可完美处理
\(\text{Code}\)
#include<cstdio>
#include<iostream>
#define LL long long
#define re register
using namespace std;
const int N = 5e6, P = 1e9 + 7;
int n, m, k, totp, pr[N], vis[N + 5], pk[N + 5];
LL sum[N + 5];
inline int fpow(LL x, LL y)
{
LL res = 1;
for(; y; y >>= 1)
{
if (y & 1) res = res * x % P;
x = x * x % P;
}
return res;
}
inline void Euler()
{
vis[1] = sum[1] = pk[1] = 1;
for(re int i = 2; i <= N; i++)
{
if (!vis[i]) pr[++totp] = i, pk[i] = fpow(i, k), sum[i] = (pk[i] - 1 + P) % P;
for(re int j = 1; j <= totp && i * pr[j] <= N; j++)
{
vis[i * pr[j]] = 1, pk[i * pr[j]] = (LL)pk[i] * pk[pr[j]] % P;
if (i % pr[j]) sum[i * pr[j]] = sum[i] * sum[pr[j]] % P;
else{sum[i * pr[j]] = sum[i] * pk[pr[j]] % P; break;}
}
}
for(re int i = 1; i <= N; i++) sum[i] = (sum[i] + sum[i - 1]) % P;
}
int main()
{
int T; scanf("%d%d", &T, &k);
Euler();
for(; T; T--)
{
scanf("%d%d", &n, &m);
LL ans = 0;
for(re int l = 1, r; l <= min(n, m); l = r + 1)
{
r = min(n / (n / l), m / (m / l));
ans = (ans + (sum[r] - sum[l - 1] + P) * (n / l) % P * (m / l)) % P;
}
printf("%lld\n", ans);
}
}