\(\color{#0066ff}{ 题目描述 }\)
给定一个正整数\(N(N\le2^{31}-1)\)
求
\(ans_1=\sum_{i=1}^n\varphi(i)\)
\(ans_2=\sum_{i=1}^n \mu(i)\)
\(\color{#0066ff}{输入格式}\)
一共T+1行 第1行为数据组数T(T<=10) 第2~T+1行每行一个非负整数N,代表一组询问
\(\color{#0066ff}{输出格式}\)
一共T行,每行两个用空格分隔的数ans1,ans2
\(\color{#0066ff}{输入样例}\)
6
1
2
8
13
30
2333
\(\color{#0066ff}{输出样例}\)
1 1
2 0
22 -2
58 -3
278 -3
1655470 2
\(\color{#0066ff}{数据范围与提示}\)
none
\(\color{#0066ff}{ 题解 }\)
可以用min_25筛写
对于\(\varphi\)
要拆成两个,一个0次项,一个1次项
在收集答案的时候直接加它们两个的差即可
对于\(\mu\)
因为只要指数超过1,就是0了,没用的,不同统计,直接统计1次的就行
#include<bits/stdc++.h>
#define LL long long
LL in() {
char ch; LL x = 0, f = 1;
while(!isdigit(ch = getchar()))(ch == '-') && (f = -f);
for(x = ch ^ 48; isdigit(ch = getchar()); x = (x << 1) + (x << 3) + (ch ^ 48));
return x * f;
}
const int maxn = 2e6 + 10;
LL g0[maxn], g1[maxn], a[maxn];
int pri[maxn];
int m, sqt, n, tot;
int getid(LL x) { return x <= sqt? x : m - n / x + 1; }
LL getphi(LL a, int b) {
if(a < pri[b]) return 0;
LL ans = (g1[getid(a)] - g1[getid(pri[b - 1])]) - (g0[getid(a)] - g0[getid(pri[b - 1])]);
for(int i = b; i <= tot && (LL)pri[i] * pri[i] <= a; i++)
for(LL x = pri[i], f = pri[i] - 1; x * pri[i] <= a; x *= pri[i], f *= pri[i])
//phi[p^2]的贡献是p*(p-1)
ans += (getphi(a / x, i + 1) * f + f * pri[i]);
return ans;
}
LL getmu(LL a, int b) {
if(a < pri[b]) return 0;
LL ans = -g0[getid(a)] + g0[getid(pri[b - 1])];
//只需枚举质数,次数就是1即可
for(int i = b; i <= tot && (LL)pri[i] * pri[i] <= a; i++)
//乘的那个f是-1,所以直接减,加的那个f是平方项=0, 不用管
ans -= getmu(a / pri[i], i + 1);
return ans;
}
int main() {
for(int T = in(); T --> 0;) {
n = in();
sqt = sqrt(n);
m = tot = 0;
for(int i = 1; i <= n; i = a[m] + 1)
a[++m] = n / (n / i), g0[m] = a[m] - 1, g1[m] = a[m] * (a[m] + 1) / 2 - 1;
for(int i = 2; i <= sqt; i++) {
if(g0[i] != g0[i - 1]) {
LL sqr = i * i;
pri[++tot] = i;
for(int j = m; a[j] >= sqr; j--) {
int id = getid(a[j] / i);
g0[j] -= g0[id] - g0[i - 1];
g1[j] -= i * (g1[id] - g1[i - 1]);
}
}
}
printf("%lld %lld\n", getphi(n, 1) + 1, getmu(n, 1) + 1);
}
return 0;
}
原文地址:https://www.cnblogs.com/olinr/p/10284336.html
时间: 2024-10-09 07:20:27