A square-free integer is an integer which is indivisible by any square number except 11. For example, 6 = 2 \cdot 36=2⋅3 is square-free, but 12 = 2^2 \cdot 312=22⋅3 is not, because 2^222 is a square number. Some integers could be decomposed into product of two square-free integers, there may be more than one decomposition ways. For example, 6 = 1\cdot 6=6 \cdot 1=2\cdot 3=3\cdot 2, n=ab6=1⋅6=6⋅1=2⋅3=3⋅2,n=ab and n=ban=ba are considered different if a \not = ba=b. f(n)f(n)is the number of decomposition ways that n=abn=ab such that aa and bb are square-free integers. The problem is calculating \sum_{i = 1}^nf(i)∑i=1nf(i).
The first line contains an integer T(T\le 20)T(T≤20), denoting the number of test cases.
For each test case, there first line has a integer n(n \le 2\cdot 10^7)n(n≤2⋅107).
For each test case, print the answer \sum_{i = 1}^n f(i)∑i=1nf(i).
\sum_{i = 1}^8 f(i)=f(1)+ \cdots +f(8)∑i=18f(i)=f(1)+⋯+f(8)
=1+2+2+1+2+4+2+0=14=1+2+2+1+2+4+2+0=14.
8 14思路：如果某个数字x拥有某一个素因子超过2个，则x的f值为0；若x的某个素因子数量为2个，则这个素因子不会对x的f值有任何的贡献；若x的某个素因子只有1个，则这个素因子贡献为2，举个例子：
60=2^2*3*5,则2没有贡献，3，5都贡献2，所以f(60)=2*2=4;
利用线性筛，每个合数只被它最小的素因子筛去，同时处理出这个数字的f值即可
AC代码：
#define _CRT_SECURE_NO_DEPRECATE #include<iostream> #include<algorithm> #include<vector> #include<cstring> #include<string> #include<cmath> using namespace std; #define INF 0x3f3f3f3f typedef unsigned long long ll; #define EPS 1e-5 const ll MOD = 1000000007; const int N_MAX =2*10000000+10; bool is_prime[N_MAX]; int prime[N_MAX],p,f[N_MAX],sum[N_MAX],number; void sieve(int n) { is_prime[0] = is_prime[1] = true; p = 0; f[1] = 1; for (int i = 2; i < n;i++) { if (!is_prime[i]) { prime[p++] = i; f[i] = 2; } for (int j = 0; j < p;j++) { number = prime[j] * i; if (number >= N_MAX)break; is_prime[number] =true; if (i%prime[j] != 0) { f[number] = f[i]<<1; } else { if (i % (prime[j] * prime[j]) == 0) { f[number] = 0; } else f[number] = f[i] >> 1; break;//线性筛，保证每个数字只被最小的素数筛去 } } } } int main() { sieve(N_MAX-5); sum[1] = 1; for (int i = 2; i < N_MAX - 9; i++) { sum[i] = sum[i - 1] + f[i]; } int t; scanf("%d",&t); while (t--) { int n; scanf("%d",&n); printf("%d\n",sum[n]); } return 0; }
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