ΣΣ [ gcd(i,j) ]

Posted 2022-07-18 tyner

https://www.luogu.org/record/22874213

题目大意:给定n和m,求Σ(1<=i<=n)Σ(1<=j<=m)GCD(i,j)*2-1

f(x) = ΣΣ[ gcd(i,j)==x ]
那么答案即为 Σf(x)x, x:1->n //规定n<m, 即gcd(n,m)<=n
g(x) = Σ[ x | gcd(i,j) ] = n/x m/x = f(x) + f(2x) + ... + f(n/x x)
i: x, 2x...n/x x, j: x, 2x...m/x x。 所以gcd(i,j)为x倍数的有 n/x m/x
移项一下，用原先预处理好的g[]算f[], f(x) = g(x) - f(2x) - f(3x) - ... - f(n/xx)
倒过来算f(x)，那么f(2x)...f(n/x x)就都算好了

#include<cstdio>
#include<iostream>
using namespace std;
#define MAX 100000+999
#define ll long long

ll f[MAX];
int n,m;

int main() 
    scanf("%d%d",&n,&m);
    if(n > m) swap(n, m);
    for(int i = 1; i <= n; i++) f[i] = (ll)(n/i) * (m/i);// 注意加括号！！！ 
    for(int i = n; i >= 1; i--) //求f[i]
        for(int j = i+i; j <= n; j+=i) 
            f[i] -= f[j];
        
    
    ll ans = 0;
    for(int i = 1; i <= n; i++) 
        ans += f[i]*i;
//  for(int i = 1; i <= n; i++) printf("%d\n",f[i]);
    printf("%lld",(ans<<1) - (ll)n*m);//把Σ变一下