[洛谷P4980]【模板】Polya定理

题目大意：给一个$n$个点的环染色，有$n$中颜色，问有多少种涂色方案是的旋转后本质不同

题解：$burnside$引理：$ans=dfrac1{|G|}sumlimits_{gin G}A_g$

对于环，有$Polya$定理：$ans=dfrac1{|G|}sumlimits_{gin G}m^{c(g)}$（$m$为颜色数，在这道题中$m=n$，$c(g)$为置换$g$中循环个数）

因为是循环相同，所以$|G|=n$，当$g=left(
egin{smallmatrix}
1&2&cdots&n-k&n-k+1&cdots&n\
k+1&k+2&cdots&n&1&cdots&k
end{smallmatrix}
ight)$时，$c(g)=gcd(k,n)$

$$
egin{align*}
ans&=dfrac1{|G|}sumlimits_{gin G}m^{c(g)}\
&=dfrac1nsumlimits_{i=1}^nn^{(i,n)}\
&=dfrac1nsumlimits_{d|n}n^dsumlimits_{i=1}^n[(i,n)=d]\
&=dfrac1nsumlimits_{d|n}n^dsumlimits_{i=1}^{lfloorfrac nd floor}[(icdot d,n)=d]\
&=dfrac1nsumlimits_{d|n}n^dsumlimits_{i=1}^{lfloorfrac nd floor}[(i,dfrac nd)=1]\
&=dfrac1nsumlimits_{d|n}n^dvarphi(dfrac nd)
end{align*}
$$

虽然是多组询问，但是依然可以$O(sqrt n)$求$varphi$，复杂度$O(Tn^{frac34})$，当然，正确的方法是求出质因数后递归求出每个因数的$varphi$，复杂度$O(Tsqrt n)$

卡点：无

C++ Code：

#include <cstdio>
const int mod = 1e9 + 7;

namespace Math {
	inline int getphi(int x) {
		int res = x;
		for (register int i = 2; i * i <= x; ++i) if (x % i == 0) {
			res = res / i * (i - 1);
			while (x % i == 0) x /= i;
		}
		if (x > 1) res = res / x * (x - 1);
		return res;
	}

	inline int pw(int base, int p) {
		static int res;
		for (res = 1; p; p >>= 1, base = static_cast<long long> (base) * base % mod) if (p & 1) res = static_cast<long long> (res) * base % mod;
		return res;
	}
	inline int inv(int x) { return pw(x, mod - 2); }
}

inline void reduce(int &x) { x += x >> 31 & mod; }

int Tim, n, ans;
inline int get(int d) {
	return static_cast<long long> (Math::pw(n, d)) * Math::getphi(n / d) % mod;
}

int main() {
	scanf("%d", &Tim);
	while (Tim --> 0) {
		scanf("%d", &n);
		ans = 0;
		for (int i = 1; i * i <= n; ++i) if (n % i == 0) {
			reduce(ans += get(i) - mod);
			if (i != n / i) reduce(ans += get(n / i) - mod);
		}
		printf("%lld
", static_cast<long long> (ans) * Math::inv(n) % mod);
	}
	return 0;
}