[AHOI2002] 芝麻开门 - 数论

求 \(n^k\) 的因子和, \(n \leq 2^{16}, k \leq 20\)

Solution

\[\prod_i \frac{p_i^{q_ik+1}-1}{p_i-1}
\]

#include <bits/stdc++.h>
using namespace std;

const int maxlen = 105;

class HP {
public:
    int len, s[maxlen];
    HP() { (*this) = 0; }
    HP(int inte) { (*this) = inte; }
    HP(const char *str) { (*this) = str; }
    friend ostream &operator<<(ostream &cout, const HP &x);
    HP operator=(int inte);
    HP operator=(const char *str);
    HP operator=(const HP &b);
    HP operator*(const HP &b);
    HP operator+(const HP &b);
    HP operator-(const HP &b);
    HP operator/(const HP &b);
    HP operator%(const HP &b);
    bool operator<(const HP &b);
    bool operator>(const HP &b);
    int Compare(const HP &b);
};

ostream &operator<<(ostream &cout, const HP &x) {
    for (int i = x.len; i >= 1; i--) cout << x.s[i];
    return cout;
}
HP HP::operator=(const char *str) {
    len = (int)strlen(str);
    for (int i = 1; i <= len; i++) s[i] = str[len - i] - '0';
    return *this;
}
HP HP::operator=(int inte) {
    if (inte == 0) {
        len = 1;
        s[1] = 0;
        return (*this);
    }
    for (len = 0; inte > 0;) {
        s[++len] = inte % 10;
        inte /= 10;
    }
    return *this;
}
HP HP::operator=(const HP &b) {
    len = b.len;
    for (int i = 1; i <= len; i++) s[i] = b.s[i];
    return *this;
}
HP HP::operator*(const HP &b) {
    int i, j;
    HP c;
    c.len = len + b.len;
    for (i = 1; i <= c.len; i++) c.s[i] = 0;
    for (i = 1; i <= len; i++)
        for (j = 1; j <= b.len; j++) c.s[i + j - 1] += s[i] * b.s[j];
    for (i = 1; i < c.len; i++) {
        c.s[i + 1] += c.s[i] / 10;
        c.s[i] %= 10;
    }
    while (c.s[i]) {
        c.s[i + 1] = c.s[i] / 10;
        c.s[i] %= 10;
        i++;
    }
    while (i > 1 && !c.s[i]) i--;
    c.len = i;
    return c;
}
HP HP::operator+(const HP &b) {
    int i;
    HP c;
    c.s[1] = 0;
    for (i = 1; i <= len || i <= b.len || c.s[i]; i++) {
        if (i <= len)
            c.s[i] += s[i];
        if (i <= b.len)
            c.s[i] += b.s[i];
        c.s[i + 1] = c.s[i] / 10;
        c.s[i] %= 10;
    }
    c.len = i - 1;
    if (c.len == 0)
        c.len = 1;
    return c;
}
HP HP::operator-(const HP &b) {
    int i, j;
    HP c;
    for (i = 1, j = 0; i <= len; i++) {
        c.s[i] = s[i] - j;
        if (i <= b.len)
            c.s[i] -= b.s[i];
        if (c.s[i] < 0) {
            j = 1;
            c.s[i] += 10;
        } else
            j = 0;
    }
    c.len = len;
    while (c.len > 1 && !c.s[c.len]) c.len--;
    return c;
}
int HP::Compare(const HP &y) {
    if (len > y.len)
        return 1;
    if (len < y.len)
        return -1;
    int i = len;
    while ((i > 1) && (s[i] == y.s[i])) i--;
    return s[i] - y.s[i];
}
bool HP::operator<(const HP &b) {
    if (len < b.len)
        return 1;
    if (len > b.len)
        return 0;
    int i = len;
    while ((i > 1) && (s[i] == b.s[i])) i--;
    return s[i] < b.s[i];
}
bool HP::operator>(const HP &b) {
    if (len > b.len)
        return 1;
    if (len < b.len)
        return 0;
    int i = len;
    while ((i > 1) && (s[i] == b.s[i])) i--;
    return s[i] > b.s[i];
}
HP HP::operator/(const HP &b) {
    int i, j;
    HP d(0), c;
    for (i = len; i > 0; i--) {
        if (!(d.len == 1 && d.s[1] == 0)) {
            for (j = d.len; j > 0; j--) d.s[j + 1] = d.s[j];
            ++d.len;
        }
        d.s[1] = s[i];
        c.s[i] = 0;
        while ((j = d.Compare(b)) >= 0) {
            d = d - b;
            c.s[i]++;
            if (j == 0)
                break;
        }
    }
    c.len = len;
    while ((c.len > 1) && (c.s[c.len] == 0)) c.len--;
    return c;
}
HP HP::operator%(const HP &b) {
    int i, j;
    HP d(0);
    for (i = len; i > 0; i--) {
        if (!(d.len == 1 && d.s[1] == 0)) {
            for (j = d.len; j > 0; j--) d.s[j + 1] = d.s[j];
            ++d.len;
        }
        d.s[1] = s[i];
        while ((j = d.Compare(b)) >= 0) {
            d = d - b;
            if (j == 0)
                break;
        }
    }
    return d;
}

int n,k,a[99],b[99],top;
HP ans;
int main() {
    cin>>n>>k;
    ans=1;
    int t=n;
    for(int i=2;i<=n;i++) {
        if(t%i==0) {
            a[++top]=i;
        }
        while(t%i==0) {
            b[top]++;
            t/=i;
        }
    }
    for(int i=1;i<=top;i++) {
        HP tmp=1,xx;
        for(int j=1;j<=b[i]*k+1;j++) xx=a[i],tmp=tmp*xx;
        xx=1;
        tmp=tmp-xx;
        xx=a[i]-1;
        tmp=tmp/xx;
        ans=ans*tmp;
    }
    cout<<ans;
}