51nod1238. 最小公倍数之和 V3(莫比乌斯反演)
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题目链接
https://www.51nod.com/Challenge/Problem.html#!#problemId=1238
题解
本来想做个杜教筛板子题结果用另一种方法过了......
所谓的“另一种方法”用到的技巧还是挺不错的,因此这里简单介绍一下。
首先还是基本的推式子:
[egin{aligned}sum_{i = 1}^n sum_{j = 1}^n { m lcm}(i, j) &= sum_{i = 1}^n sum_{j = 1}^n frac{ij}{{ m gcd}(i, j)} \ &= sum_{d = 1}^{n} sum_{i = 1}^n sum_{j = 1}^n[{ m gcd}(i, j) = d]frac{ij}{d} \ &= sum_{d = 1}^n dsum_{i = 1}^{leftlfloorfrac{n}{d} ight floor}sum_{j = 1}^{leftlfloorfrac{n}{d} ight floor}[{ m gcd}(i, j) = 1]ijend{aligned}]
设 (f(x) = sum_limits{i = 1}^x sum_limits{j = 1}^x [{ m gcd}(i, j) = 1]ij),那么答案即为 (sum_limits{d = 1}^n d imes f(leftlfloorfrac{n}{d} ight floor))。显然答案可以数论分块求,因此我们的任务就是求 (f) 函数。
首先考虑当 (x) 较小时,我们能否直接预处理出 (f(x))。考虑差分:当 (x > 1) 时,(f(x)) 较 (f(x - 1)) 而言,多了的部分为:[left(sum_limits{i = 1}^xsum_limits{j = 1}^x [{ m gcd}(i, j) = 1]ij ight) - left(sum_limits{i = 1}^{x - 1}sum_limits{j = 1}^{x - 1} [{ m gcd}(i, j) = 1]ij ight) = 2x sum_{i = 1}^x [{ m gcd}(i, x) = 1]i]
而由于小于等于 (x(x > 1)) 且与 (x) 互质的数的和为 (frac{varphi(x)x}{2})(证明提示:当 (n geq 2) 时,若 ({ m gcd}(d, n) = 1) 必然有 ({ m gcd}(n - d, n) = 1),与 (n) 互质的 (d) 共有 (varphi(n)) 个),因此我们就得到了 (f(x)) 的递推式:(f(x) = f(x - 1) + 2x imes frac{varphi(x)x}{2}),即 (f(x) = f(x - 1) + varphi(x)x^2)。
不过这只能处理 (x) 较小的情况。当 (x) 较大时,我们仍然得另谋他路。在 (f(x) = sum_limits{i = 1}^x sum_limits{j = 1}^x [{ m gcd}(i, j) = 1]ij) 当中,对 (f(x)) 有贡献的 (i, j) 满足 (i) 与 (j) 是互质的,我们考虑补集转化,用总和减去不互质的 (i, j) 的贡献:
[egin{aligned} f(x) &= sum_{i = 1}^xsum_{j = 1}^x ij - sum_{d = 2}^x sum_{i = 1}^x sum_{j = 1}^x [{ m gcd}(i, j) = d]ij \ &= left(frac{x(x+1)}{2} ight)^2 - sum_{d = 2}^x d^2 sum_{i = 1}^{leftlfloorfrac{x}{d} ight floor} sum_{j = 1}^{leftlfloorfrac{x}{d} ight floor} [{ m gcd}(i, j) = 1]ij \ &=left(frac{x(x+1)}{2} ight)^2 - sum_{d = 2}^x d^2 f(leftlfloorfrac{x}{d} ight floor) end{aligned}]
这样,我们就能够递归地去求解当 (x) 较大时 (f(x)) 的值了。不难发现,该求解方法的时间复杂度和杜教筛是一样的,为 (O(n^{frac{2}{3}})),且非常好写。
代码
#include<bits/stdc++.h>
using namespace std;
const int mod = 1000000007, inv2 = 500000004, inv6 = 166666668, up = 10000001;
int main() {
function<int (int, int)> mul = [&] (int x, int y) {
return (long long) x * y % mod;
};
function<void (int&, int)> add = [&] (int& x, int y) {
x += y;
if (x >= mod) {
x -= mod;
}
};
function<void (int&, int)> sub = [&] (int& x, int y) {
x -= y;
if (x < 0) {
x += mod;
}
};
vector<bool> is_prime(up, true);
vector<int> phi(up), primes;
phi[1] = 1;
for (int i = 2; i < up; ++i) {
if (is_prime[i]) {
primes.push_back(i);
phi[i] = i - 1;
}
for (auto v : primes) {
int d = v * i;
if (d >= up) {
break;
}
is_prime[d] = false;
if (i % v == 0) {
phi[d] = mul(phi[i], v);
break;
} else {
phi[d] = mul(phi[i], phi[v]);
}
}
}
for (int i = 2; i < up; ++i) {
phi[i] = mul(mul(phi[i], i), i);
add(phi[i], phi[i - 1]);
}
function<int (long long)> sum_pow2 = [&] (long long n) {
n %= mod;
return mul(mul(mul(n, n + 1), (n * 2 + 1)), inv6);
};
map<long long, int> value;
function<int (long long)> f = [&] (long long n) {
if (value.count(n)) {
return value[n];
} else {
int result = 0;
if (n < up) {
result = phi[n];
} else {
int x = n % mod;
x = mul(mul(x, x + 1), inv2);
result = mul(x, x);
for (long long i = 2, last; i <= n; i = last + 1) {
last = n / (n / i);
sub(result, mul((sum_pow2(last) - sum_pow2(i - 1) + mod) % mod, f(n / i)));
}
}
return value[n] = result;
}
};
long long n;
scanf("%lld", &n);
int answer = 0;
for (long long i = 1, last; i <= n; i = last + 1) {
last = n / (n / i);
add(answer, mul(mul(mul((i + last) % mod, (last - i + 1) % mod), inv2), f(n / i)));
}
printf("%d
", answer);
return 0;
}
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