LOJ572: Misaka Network 与求和

传送门

假设 \(f^k(i)\) 就是 \(f(i)\)

莫比乌斯反演得到

\[ans=\sum_{i=1}^{N}\lfloor\frac{N}{i}\rfloor^2\sum_{d|i}f(d)\mu(\frac{i}{d})
\]

令 \(g(N)=\sum_{i=1}^{N}(f\times \mu)(i)\)

而 \((f\times \mu)\times 1=f\times (\mu\times 1)=f\)

所以

\[\sum_{i=1}^{N}f(i)=\sum_{i=1}^{N}(f\times \mu \times 1)(i)=\sum_{i=1}^{N}g(\lfloor\frac{N}{i}\rfloor)
\]

\[g(N)=\sum_{i=1}^{N}f(i)-\sum_{i=2}^{N}g(\lfloor\frac{N}{i}\rfloor)
\]

类似 \(UOJ188:sanrd\) 一样筛出 \(f\) 的和即可

# include <bits/stdc++.h>
using namespace std;
typedef long long ll;
typedef unsigned int uint;

const int maxn(1e6 + 5);

inline uint Pow(uint x, int y) {
	register uint ret = 1;
	for (; y; y >>= 1, x = x * x) if (y & 1) ret = ret * x;
	return ret;
}

int pr[maxn], tot, id1[maxn], id2[maxn], d, cnt, k;
bitset <maxn> ispr;
uint n, f[maxn], val[maxn], prk[maxn], g[maxn];

inline void Sieve(int mx) {
	register int i, j;
	for (i = 2, ispr[1] = 1; i <= mx; ++i) {
		if (!ispr[i]) pr[++tot] = i, prk[tot] = Pow(i, k);
		for (j = 1; j <= tot && pr[j] * i <= mx; ++j) {
			ispr[pr[j] * i] = 1;
			if (!(i % pr[j])) break;
		}
	}
}

# define ID(x) (x) <= d ? id1[x] : id2[n / (x)]

uint Calc(uint x, int m) {
	if (x <= 1 || pr[m] > x) return 0;
	register uint i, t, ret = 0;
	for (i = m; i <= tot && (ll)pr[i] * pr[i] <= x; ++i)
		for (t = pr[i]; (ll)pr[i] * t <= x; t *= pr[i])
			ret += Calc(x / t, i + 1) + (f[ID(x / t)] - i + 1) * prk[i];
	return ret;
}

inline void Init(uint _n) {
	register uint i, j;
	for (cnt = 0, d = sqrt(n = _n), i = 1; i <= n; i = j + 1) {
		j = n / (n / i), val[++cnt] = n / i;
		val[cnt] <= d ? id1[val[cnt]] = cnt : id2[n / val[cnt]] = cnt;
		f[cnt] = val[cnt] - 1;
	}
	for (i = 1; i <= tot && (ll)pr[i] * pr[i] <= n; ++i)
		for (j = 1; j <= cnt && (ll)pr[i] * pr[i] <= val[j]; ++j)
			f[j] -= f[ID(val[j] / pr[i])] - i + 1;
}

inline uint Solve(uint r) {
	if (~g[ID(r)]) return g[ID(r)];
	register uint ret = Calc(r, 1) + f[ID(r)], i, j;
	for (i = 2, j; i <= r; i = j + 1) j = r / (r / i), ret -= Solve(r / i) * (j - i + 1);
	return g[ID(r)] = ret;
}

int main() {
	memset(g, -1, sizeof(g));
	scanf("%u%d", &n, &k), Sieve(sqrt(n)), Init(n);
	register uint i, j, ans = 0, lst = 0, cur;
	for (i = 1; i <= n; i = j + 1) {
		j = n / (n / i), cur = Solve(j);
		ans += (n / i) * (n / i) * (cur - lst);
		lst = cur;
	}
	printf("%u\n", ans);
	return 0;
}