POJ 1845 Sumdiv 快速求幂+同余+乘法逆元

题意：给定A, B，求A^B的所有因数之和，并模9901。

题解：

1: 对A进行素因子分解得

     A = p1^a1 * p2^a2 * p3^a3 *...* pn^an.
  故 A^B = p1^(a1*B) * p2^(a2*B) *...* pn^(an*B);

2：A^B的所有约数之和为：

     sum = [1+p1+p1^2+...+p1^(a1*B)] * [1+p2+p2^2+...+p2^(a2*B)] *...* [1+pn+pn^2+...+pn^(an*B)].

  如 200 = 2^3 * 5^2 :  sum(200) = [1 + 2 + 4 + 8] * [1 + 5 + 25].

3：等比求和公式 (p^(n+1)-1)/(p-1)，由于这里有1/(p-1)，需要求(p-1)关于mod的逆元（若ax=1 mod m 则称a关于模f的乘法逆元为x。也可表示为ax≡1(mod m)。） 但是，存在逆元需要 gcd(a,m)  = 1。所以我们直接二分+快速求幂。

#include<cstdio>
#include<cmath>
#include<cstring>
#include<algorithm>
using namespace std;

#define lint __int64
#define MAXN 100000
#define M 9901
struct Factor { lint base, exp; };
Factor f[MAXN]; lint fn;
lint p[MAXN], a[MAXN], pn;

void prime ()
{
    int i, j; pn = 0;
    memset(a,0,sizeof(a));
    for ( i = 2; i < MAXN; i++ )
    {
        if ( !a[i] ) p[pn++] = i;
        for ( j = 0; j < pn && i * p[j] < MAXN && (p[j] <= a[i] || a[i] == 0); j++ )
            a[i*p[j]] = p[j];
    }
}

void Factorization ( int num )
{
    fn = 0;
    for ( int i = 0; i < pn && p[i] <= num; i++ )
    {
        if ( num % p[i] ) continue;
        f[++fn].base = p[i];
        f[fn].exp = 0;
        while ( num % p[i] == 0 )
        {
            f[fn].exp++;
            num /= p[i];
        }
    }
    if ( num != 1 )
    {
        f[++fn].base = num;
        f[fn].exp = 1;
    }
}

int mod_exp ( int a, lint b )
{
    int ret = 1;
    a = a % M;
    while ( b >= 1 )
    {
        if ( b & 1 ) ret = ret * a % M;
        a = a * a % M;
        b >>= 1;
    }
    return ret;
}

int sum_exp ( int p, lint exp )
{
    if ( exp == 0 ) return 1;
    lint tmp1, tmp2, mid = exp / 2;
    if ( exp & 1 )
    {
        tmp1 = sum_exp (p, mid);
        tmp2 = mod_exp (p, mid + 1);
        return (tmp1+tmp2*tmp1) % M;
    }
    else
    {
        tmp1 = sum_exp (p, mid-1);
        tmp2 = mod_exp (p, mid);
        return (tmp1 + tmp2 + p*tmp2*tmp1) % M;
    }
}

int main()
{
    prime();
    int A, B, ret = 1;
    scanf("%d%d",&A,&B);
    if ( A == 0 ) {printf("0\n");return 0;}
    if ( B == 0 || A == 1 ) {printf("1\n"); return 0;}
    Factorization ( A );
    for ( int i = 1; i <= fn; i++ )
        ret = ret * sum_exp(f[i].base, f[i].exp*B) % M;
    printf("%d\n",ret);
}