poj 2480 Longge's problem(欧拉函数)
Description
Longge is good at mathematics and he likes to think about hard mathematical problems which will be solved by some graceful algorithms. Now a problem comes: Given an integer N(1 < N < 2^31),you are to calculate ∑gcd(i, N) 1<=i <=N.
"Oh, I know, I know!" Longge shouts! But do you know? Please solve it.
"Oh, I know, I know!" Longge shouts! But do you know? Please solve it.
Input
Input contain several test case.
A number N per line.
A number N per line.
Output
For each N, output ,∑gcd(i, N) 1<=i <=N, a line
Sample Input
2 6
Sample Output
3 15
解题思路:给出一个数n,求1-n这n个数与n的最大公约数之和。举个栗子:当n=4时,1,2,3,4与4的最大公约数分别为1,2,1,4,累加和为8。正解:1-n中每个数与n的最大公约数肯定是n的一个因子,所以我们只需要枚举n的每一个因子x∈[1,√n],然后看有多少个满足gcd(k,n)==x,即求满足gcd(k/x,n/x)==1中k的个数(用欧拉函数求解),则公式为:∑x*[gcd(k/x,n/x)==1]。
AC代码(204ms):
1 #include <iostream> 2 #include <cstdio> 3 #include <cstring> 4 #include <map> 5 #include <vector> 6 #include <set> 7 using namespace std; 8 typedef long long LL; 9 const int maxn = 1e6+5; 10 LL n, ans; 11 LL get_Euler(LL x){ 12 LL res = x; 13 for(LL i = 2LL; i * i <= x; ++i) { 14 if(x % i == 0) { 15 res = res / i * (i - 1); 16 while(x % i == 0) x /= i; 17 } 18 } 19 if(x > 1LL) res = res / x * (x - 1); 20 return res; 21 } 22 23 int main(){ 24 while(cin >> n) { 25 ans = 0LL; 26 for (LL i = 1LL; i * i <= n; ++i) { 27 if(n % i == 0) { 28 ans += i * get_Euler(n / i); 29 if(i * i != n) ans += n / i * get_Euler(i); ///避免重复计数 30 } 31 } 32 cout << ans << endl; 33 } 34 return 0; 35 }
AC代码二(32ms):思路和上面相同,只是将问题求解转换一下gcd(i, n) == (p_i)^j,即求Σ(p_i)^j [gcd(i/((p_i)^j)), n/((p_i)^j)==1],化简公式得 (k+1)* p^k - k*p^(k-1),再根据积性函数的性质得n的欧拉函数值为每种素因子对应的欧拉函数值φ((p_i)^a_i)相乘即可。时间复杂度是O(sqrt(n))。具体推导过程:传送门
1 #include <iostream> 2 #include <cstdio> 3 #include <cstring> 4 using namespace std; 5 typedef long long LL; 6 LL n; 7 LL solve(LL x) { 8 LL p_i, k, ans = 1LL; 9 for(LL i = 2LL; i * i <= x; ++i) { 10 if(x % i == 0) { 11 p_i = 1LL, k = 0; 12 while(x % i == 0) {k++, p_i *= i, x /= i;} 13 ans *= (k + 1) * p_i - k * p_i / i; ///(k+1)*p^k - k*p^(k-1) 14 } 15 } 16 if(x > 1LL) ans *= 2 * x - 1LL; 17 return ans; 18 } 19 int main() { 20 while(cin >> n) { 21 cout << solve(n) << endl; 22 } 23 return 0; 24 }