文章目錄

• title
• solution
• code

title

solution

這道題是多么的妙啊，完全不是我能推出來的式子呢！

觀察數據范圍，有點奇怪欸，在暗示我？？

考慮暴力枚舉$n$
$$S(n,m)=\sum _{i=1}^m\phi (n\times i)$$
神奇的操作來了，將$n$質因數分解，并把不同的質因數分別拿出一個
$n=\prod p_i^{e_i}$
$q=\prod p_i$
$p=\prod p_i^{e_i?1}$
則有$p\times q=n$

若$i\%j=0$，則$\phi (ij)=\phi (i)\times j$

若$(i,j)=1$，則$\phi (ij)=\phi (i)\times \phi (j)$

$$S(n,m)=\sum _{i=1}^m\phi (n\times i)$$$$=p?\sum _{i=1}^m\phi (q\times i)$$$$=p?\sum _{i=1}^m\phi (\frac{q}{gcd(q,i)}\times i\times gcd(q,i))$$$$=p?\sum _{i=1}^m\phi (\frac{q}{gcd(q,i)})\phi (i\times gcd(q,i))$$$$=p?\sum _{i=1}^m\phi (\frac{q}{gcd(q,i)})\phi (i)gcd(q,i)$$$$=p\sum _{i=1}^m\phi (\frac{q}{gcd(q,i)})\phi (i)\sum _{d|gcd(q,i)}\phi (d)$$$$=p\sum _{i=1}^m\phi (i)\sum _{d|i,d|q}\phi (\frac{q}{d})$$$$=p\sum _{d|q}\phi (\frac{q}{d})\sum _{i=1}^{?\frac{m}{d}?}\phi (i\times d)$$$$=p\sum _{d|q}\phi (\frac{q}{d})S(d,?\frac{m}{d}?)$$

$\phi$用杜教篩，應該是老熟人了
$S(n,m)$記憶化一下，應該就沒了

code

#include <cstdio> #include <vector> #include <map> using namespace std; #define mod 1000000007 #define int long long #define maxn 200000 map < int, int > mp, s[maxn]; int cnt; int minp[maxn + 5]; //minp[i]:i的最大質因子 int prime[maxn], phi[maxn + 5]; //phi[i]:1~i的phi的前綴和 bool vis[maxn + 5];void init() {phi[1] = 1;for( int i = 2;i <= maxn;i ++ ) {if( ! vis[i] ) prime[++ cnt] = i, minp[i] = i, phi[i] = i - 1;for( int j = 1;j <= cnt && i * prime[j] <= maxn;j ++ ) {vis[i * prime[j]] = 1, minp[i * prime[j]] = prime[j];if( i % prime[j] == 0 ) {phi[i * prime[j]] = phi[i] * prime[j] % mod;//與式子推導的第二步為什么p能直接從φ里面拿出來呼應break;}elsephi[i * prime[j]] = phi[i] * ( prime[j] - 1 ) % mod;}}for( int i = 1;i <= maxn;i ++ ) phi[i] = ( phi[i] + phi[i - 1] ) % mod; }int Phi( int n ) {if( n <= maxn ) return phi[n];if( mp[n] ) return mp[n];int ans = n * ( n + 1 ) / 2 % mod;for( int i = 2, r;i <= n;i = r + 1 ) {r = n / ( n / i );ans = ( ans - ( r - i + 1 ) * Phi( n / i ) % mod + mod ) % mod;}return mp[n] = ans; }int solve( int n, int m ) {if( ! m ) return 0;if( s[n][m] ) return s[n][m];if( n == 1 ) return s[n][m] = Phi( m );if( m == 1 ) return s[n][m] = ( Phi( n ) - Phi( n - 1 ) + mod ) % mod;vector < int > g;int p = 1, q = 1, N = n, x;while( N > 1 ) {x = minp[N], q *= x, N /= x, g.push_back( x );while( N % x == 0 ) N /= x, p *= x;}int len = g.size(), ans = 0;for( int i = 0;i < ( 1 << len );i ++ ) { //枚舉q的所有質因子(狀壓) int d = 1;for( int j = 0;j < len;j ++ )if( i & ( 1 << j ) ) d = d * g[j]; //二進制位為1則有該質因子ans = ( ans + ( Phi( q / d ) - Phi( q / d - 1 ) + mod ) % mod * solve( d, m / d ) % mod ) % mod;}return s[n][m] = ans * p % mod; }signed main() {int n, m;scanf( "%lld %lld", &n, &m );init();int ans = 0;for( int i = 1;i <= n;i ++ ) ans = ( ans + solve( i, m ) ) % mod;printf( "%lld\n", ans );return 0; }

總結

以上是生活随笔為你收集整理的DZY Loves Math IV（杜教筛）的全部內容，希望文章能夠幫你解決所遇到的問題。

如果覺得生活随笔網站內容還不錯，歡迎將生活随笔推薦給好友。