diff --git a/TangDou/Topic/SpareTire.cpp b/TangDou/Topic/SpareTire.cpp index 15adefd..4ee8081 100644 --- a/TangDou/Topic/SpareTire.cpp +++ b/TangDou/Topic/SpareTire.cpp @@ -3,18 +3,18 @@ using namespace std; #define int long long #define endl "\n" -const int Six = 166666668; // 6,2关于mod的乘法逆元 -const int Two = 500000004; const int mod = 1e9 + 7; // 尽量这样定义mod ,减少非必要的麻烦 -int Mod(int a, int b) { - return (a % mod) * (b % mod) % mod; -} - -// 递推函数可以用通项式直接求解 -// 通项式:a[kn]=(k*n)^2+k*n -int F(int k, int n) { - return (Mod(k, k) * Mod(Mod(n, n + 1), Mod(n + n + 1, Six)) % mod + Mod(Mod(1 + n, n), Mod(k, Two))) % mod; +// 快速幂 +int qmi(int a, int b) { + int res = 1; + a %= mod; + while (b) { + if (b & 1) res = res * a % mod; + b >>= 1; + a = a * a % mod; + } + return res; } vector p; // 将m拆分成的质数因子序列p @@ -25,11 +25,17 @@ signed main() { #endif int n, m; + int rev6 = qmi(6, mod - 2); + int rev2 = qmi(2, mod - 2); + while (cin >> n >> m) { - int sum = F(1, n), ans = 0; // 计算总和sum + int res = n * (n + 1) % mod * (2 * n + 1) % mod * rev6 % mod; + int t = n * (n + 1) % mod * rev2 % mod; + + res = (res + t) % mod; // 质因子分解 - int t = m; // 复制出来 + t = m; // 复制出来 for (int i = 2; i * i <= t; i++) { if (t % i == 0) { p.push_back(i); @@ -45,19 +51,23 @@ signed main() { 例如i=3(11二进制) 表示取前两个因子,i=5(101)表示取第1个和第3个的因子 */ for (int i = 1; i < (1 << p.size()); i++) { - int cnt = 0, s = 1; + int cnt = 0, tmp = 1; for (int j = 0; j < p.size(); j++) if ((i >> j) & 1) { cnt++; - s *= p[j]; + tmp *= p[j]; } + int nn = n / tmp; + int pt = (nn) % mod * (nn + 1) % mod * (2 * nn + 1) % mod * rev6 % mod; + pt = pt * tmp % mod * tmp % mod; + pt = (pt + nn * (tmp + tmp * nn) % mod * rev2 % mod) % mod; if (cnt & 1) - ans = (ans + F(s, n / s)) % mod; // 根据容斥,取奇数个因子时,应加上 + res = (res - pt + mod) % mod; else - ans = ((ans - F(s, n / s)) % mod + mod) % mod; // 偶数的减,减法需要防负数 + res = (res + pt) % mod; } // 补集,标准取模动作 - cout << ((sum - ans) % mod + mod) % mod << endl; + cout << res << endl; } }