2020 Multi-University Training Contest 6 - 1007. A Very Easy Math Problem(莫比乌斯反演)

题目链接：A Very Easy Math Problem

题意：给你一个T，x，k，表示有T次询问，每次询问给你一个n，求：

$$\sum_{a_1=1}^{n}\sum_{a_2=1}^{n}\ldots \sum_{a_x=1}^{n}\left (\prod_{j=1}^{x}a_j^k\right )f(\gcd(a_1,a_2,\ldots ,a_x)) * \gcd(a_1,a_2,\ldots ,a_x)$$

其中，如果x有平方因子，则f(x)=0，否则f(x)=1，$T\leq 10^4,1\leq k\leq 10^9,1\leq x\leq 10^9,1\leq n\leq 2\times 10^5$

思路：先考虑一下f(x)的求法，将x质因子分解后，如果某一个质数的幂次$\geq 2$，那么x一定有平方因子，那么f(x)=0，并且由莫比乌斯函数的定义有$\mu(x)=0$，当f(x)=1时，$\mu(x)=1$或者$\mu(x)=-1$，所以可以得到$f(x)={\mu(x)}^2$

$$\sum_{a_1=1}^{n}\sum_{a_2=1}^{n}\ldots \sum_{a_x=1}^{n}\left (\prod_{j=1}^{x}a_j^k\right )f(\gcd(a_1,a_2,\ldots ,a_x)) * \gcd(a_1,a_2,\ldots ,a_x)$$

令$\gcd(a_1,a_2,\ldots ,a_x)=d$

$$\sum_{a_1=1}^{n}\sum_{a_2=1}^{n}\ldots \sum_{a_x=1}^{n}\left (\prod_{j=1}^{x}a_j^k\right )f(d) * d [\gcd(a_1,a_2,\ldots ,a_x)==d]$$

把d提到最前面

$$\sum_{d=1}^{n} \sum_{a_1=1}^{\frac{n}{d}}\sum_{a_2=1}^{\frac{n}{d}}\ldots \sum_{a_x=1}^{\frac{n}{d}}(a_1 \cdots a_x)^k * f(d) * d * d^{kx} [\gcd(a_1,a_2,\ldots ,a_x)==1]$$

运用反演公式

$$\sum_{d=1}^{n} \sum_{a_1=1}^{\frac{n}{d}}\sum_{a_2=1}^{\frac{n}{d}}\ldots \sum_{a_x=1}^{\frac{n}{d}}(a_1 \cdots a_x)^k * f(d) * d * d^{kx} * \sum_{j \mid gcd} \mu(j)$$

同样的，把j提到前面

$$\sum_{d=1}^{n} \sum_{j=1}^{\frac{n}{d}} \sum_{a_1=1}^{\frac{n}{dj}}\sum_{a_2=1}^{\frac{n}{dj}}\ldots \sum_{a_x=1}^{\frac{n}{dj}}(a_1 \cdots a_x)^k * f(d) * d * d^{kx} * j^{kx} * \mu(j)$$

由于$\sum_{a_1=1}^{\frac{n}{dj}}\sum_{a_2=1}^{\frac{n}{dj}}\ldots \sum_{a_x=1}^{\frac{n}{dj}}(a_1 \cdots a_x)^k={[1^k+2^k+\cdots + (\frac{n}{dj})^k]}^x$

$$\sum_{d=1}^{n} \sum_{j=1}^{\frac{n}{d}} \mu(j) * (dj)^{kx} * f(d) * d * {[1^k+2^k+\cdots + (\frac{n}{dj})^k]}^x$$

令T=dj

$$\sum_{T=1}^n \sum_{j \mid T} \mu(j) * T^{kx} * f(\frac{T}{j}) * \frac{T}{j} * {[1^k+2^k+\cdots + (\frac{n}{dj})^k]}^x$$

令$g(T)=\sum_{j \mid T} \mu(j) * T^{kx} * f(\frac{T}{j}) * \frac{T}{j}$

枚举j的倍数预处理g(T)，直接预处理${[1^k+2^k+\cdots + (\frac{n}{dj})^k]}^x$

每次询问的时候对n数论分块，时间复杂度$T\sqrt n+nlogn$

#include <iostream>
#include <algorithm>
#include <cstring>
#include <cstdio>

using namespace std;

typedef long long ll;

const int N = 200010;
const ll mod = 1000000007;

int T, n;
int tot, isp[N], p[N], mu[N], f[N];
ll k, x, s[N], qmi[N], g[N];

void init(int n)
{
    mu[1] = 1;
    for (int i = 2; i <= n; i++) {
        if (!isp[i]) {
            p[++tot] = i;
            mu[i] = -1;
        }
        for (int j = 1; j <= tot && p[j] <= n / i; j++) {
            isp[i * p[j]] = 1;
            if (0 == i % p[j]) {
                mu[i * p[j]] = 0;
                break;
            }
            else mu[i * p[j]] = -mu[i];
        }
    }
    for (int i = 1; i <= n; i++) f[i] = mu[i] * mu[i];
}

ll power(ll a, ll n)
{
    ll res = 1;
    while (n) {
        if (n & 1) res = res * a % mod;
        a = a * a % mod;
        n >>= 1;
    }
    return res;
}

int main()
{
    // freopen("in.txt", "r", stdin);
    // freopen("out.txt", "w", stdout);
    init(N - 5);
    scanf("%d%lld%lld", &T, &k, &x);
    for (int i = 0; i < N; i++) {
        qmi[i] = power(i, x * k % (mod - 1));
        s[i] = power(i, k);
    }
    for (int i = 1; i < N; i++) s[i] = (s[i] + s[i - 1]) % mod;
    for (int i = 0; i < N; i++) s[i] = power(s[i], x);
    for (int i = 1; i <= N - 5; i++) {
        for (int j = 1; j <= (N - 5) / i; j++) {
            int t = i * j;
            ll tv = qmi[t] * mu[j] % mod * f[i] % mod * i % mod;
            g[t] = ((g[t] + tv) % mod + mod) % mod;
        }
    }
    for (int i = 1; i <= N - 5; i++) g[i] = (g[i] + g[i - 1]) % mod;
    while (T--) {
        scanf("%d", &n);
        ll res = 0;
        for (int l = 1, r = 0; l <= n; l = r + 1) {
            r = n / (n / l);
            ll t = ((g[r] - g[l - 1]) % mod + mod) % mod;
            t = t * s[n / l] % mod;
            res = (res + t) % mod;
        }
        printf("%lld\n", res);
    }
    return 0;
}