E. Number Challenge

推式子

$$\begin{gathered} \sum _{i=1}^a\sum _{j=1}^b\sum _{k=1}^c\sigma (ijk) \\ =\sum _{i=1}^a\sum _{j=1}^b\sum _{k=1}^c\sum _{x\mid i}\sum _{y\mid j}\sum _{z\mid k}(gcd(x,y)=1)(gcd(x,z)=1)(gcd(y,z)=1) \\ =\sum _{x=1}^a\sum _{y=1}^b\sum _{z=1}^c\lfloor \frac{a}{x}\rfloor \lfloor \frac{b}{y}\rfloor \lfloor \frac{c}{z}\rfloor (gcd(x,y)=1)(gcd(x,z)=1)(gcd(y,z)=1) \\ =\sum _{d=1}^a\mu (d)\sum _{x=1}^{\lfloor \frac{a}{d}\rfloor }\lfloor \frac{a}{dx}\rfloor \sum _{y=1}^{\lfloor \frac{b}{d}\rfloor }\lfloor \frac{b}{dy}\rfloor \sum _{z=1}^c\lfloor \frac{c}{d}\rfloor (gcd(x,z)=1)(gcd(y,z)=1) \\ =\sum _{d=1}^a\mu (d)\sum _{z=1}^c\lfloor \frac{c}{d}\rfloor \sum _{x=1}^{\lfloor \frac{a}{d}\rfloor }\lfloor \frac{a}{dx}\rfloor (gcd(x,z)==1)\sum _{y=1}^{\lfloor \frac{b}{d}\rfloor }\lfloor \frac{b}{dx}\rfloor (gcd(y,z)==1) \end{gathered}$$

然后就是的$n^2\log n$乱搞了。

代码

/*Author : lifehappy
*/
#pragma GCC optimize(2)
#pragma GCC optimize(3)
#include <bits/stdc++.h>#define mp make_pair
#define pb push_back
#define endl '\n'
#define mid (l + r >> 1)
#define lson rt << 1, l, mid
#define rson rt << 1 | 1, mid + 1, r
#define ls rt << 1
#define rs rt << 1 | 1using namespace std;typedef long long ll;
typedef unsigned long long ull;
typedef pair<int, int> pii;const double pi = acos(-1.0);
const double eps = 1e-7;
const int inf = 0x3f3f3f3f;inline ll read() {ll f = 1, x = 0;char c = getchar();while(c < '0' || c > '9') {if(c == '-')    f = -1;c = getchar();}while(c >= '0' && c <= '9') {x = (x << 1) + (x << 3) + (c ^ 48);c = getchar();}return f * x;
}const int N = 2e3 + 10, mod = 1073741824;int g[N][N], prime[N], cnt;bool st[N];ll mu[N], a, b, c;void init() {mu[1] = 1;for(int i = 2; i < N; i++) {if(!st[i]) {prime[cnt++] = i;mu[i] = -1;}for(int j = 0; j < cnt && i * prime[j] < N; j++) {st[i * prime[j]] = 1;if(i % prime[j] == 0) break;mu[i * prime[j]] = -mu[i];}}
}int gcd(int a, int b) {if(!b) return a;if(g[a][b]) return g[a][b];return g[a][b] = gcd(b, a % b);
}void get_gcd() {for(int i = 1; i <= c; i++) {for(int j = 1; j <= c; j++) {g[i][j] = gcd(i, j);}}
}void Sort(ll & a, ll & b, ll & c) {ll A = a, B = b, C = c;a = min({A, B, C});c = max({A, B, C});b = A + B + C - a - c;
}ll f(ll n, ll m) {ll ans = 0;for(int i = 1; i <= n; i++) {if(g[i][m] == 1) {ans += n / i;}}ans = (ans + mod) % mod;return ans;
}int main() {// freopen("in.txt", "r", stdin);// freopen("out.txt", "w", stdout);// ios::sync_with_stdio(false), cin.tie(0), cout.tie(0);a = read(), b = read(), c = read();Sort(a, b, c);init();get_gcd();ll ans = 0;for(int i = 1; i <= a; i++) {for(int j = 1; j <= b; j++) {if(g[i][j] == 1) {ans = (ans + mu[j] * (a / i) % mod * f(b / j, i) % mod * f(c / j, i) % mod + mod) % mod;}}}cout << ans << endl;return 0;
}