分治 递归
点击查看代码
#include<bits/stdc++.h>
using namespace std;
typedef long long LL;
const int N = 1e5 + 10;
const LL mod = 9901;
LL a, b;
LL qmi(LL a, LL k)
{
LL res = 1;
while (k) {
if (k & 1)
res = res * a % mod;
k >>= 1;
a = a * a % mod;
}
return res;
}
LL sum(LL p, LL k)
{
if (k == 1)
return 1;
if (k % 2)
return (sum(p, k -1) + qmi(p, k - 1)) % mod;
else
return (1 + qmi(p, k / 2)) * sum(p, k / 2) % mod;
}
void solve()
{
cin >> a >> b;
LL res = 1;
for (LL i = 2; i <= a / i; i ++) {
if (a % i == 0) {
LL cnt = 0;
while (a % i == 0) {
a /= i;
cnt ++;
}
res = res * sum(i, cnt * b + 1) % mod;
}
}
if (a > 1)
res = res * sum(a, b + 1) % mod;
if (a == 0)
res = 0;
cout << res << endl;
}
signed main()
{
ios::sync_with_stdio(false);
cin.tie(nullptr);
cout.tie(nullptr);
solve();
return 0;
}
- 约数之和
由算数基本定理可知,\(A = P_1^{\alpha_1} \cdot P_2^{\alpha_2} \cdot \cdots \cdot P_k^{\alpha_k}\),那么 \(A^{B} = P_1^{\alpha_1 \cdot B} \cdot P_2^{\alpha_2 \cdot B} \cdot \cdots \cdot P_k^{\alpha_k \cdot B}\)
对于 \(A\),约数之和为 \((P_1^{0} + P_1^{1} + \cdots P_1^{\alpha_1}) \cdot (P_2^{0} + P_2^{1} + \cdots P_2^{\alpha_2}) \cdots (P_k^{0} + P_k^{1} + \cdots P_k^{\alpha_k})\)
对于 \(A^{B}\),约数之和为 \((P_1^{0} + P_1^{1} + \cdots P_1^{\alpha_1 \cdot B}) \cdot (P_2^{0} + P_2^{1} + \cdots P_2^{\alpha_2 \cdot B}) \cdots (P_k^{0} + P_k^{1} + \cdots P_k^{\alpha_k \cdot B})\)
- 公式推导
定义 \(sum(P, k) = P^{0} + P^{1} + \cdots + P^{k - 1}\)
① 当 \(k\) 是偶数时
\(sum(P, k) = P^{0} + P^{1} + \cdots + P^{\frac{k}{2} - 1} + P^{\frac{k}{2}} + P^{\frac{k}{2} + 1} + \cdots + P^{k - 1} = P^{0} + P^{1} + \cdots + P^{\frac{k}{2} - 1} + P^{\frac{k}{2}} \cdot (P^{0} + P^{1} + \cdots + P^{\frac{k}{2} - 1}) = sum(P, \frac{k}{2}) + P^{\frac{k}{2}} \cdot sum(P, \frac{k}{2}) = (1 + P^{\frac{k}{2}}) \cdot sum(P, \frac{k}{2})\)
② 当 \(k\) 是奇数时
\(sum(P, k) = P^{0} + P^{1} + \cdots + P^{k - 2} + P^{k - 1} = sum(P, k - 1) + P^{k - 1}\),\(k - 1\) 是偶数,可以用偶数的递推公式
- 特殊情况
当 \(a = 0\) 时,约数之和为 \(0\)