「QuestOJ」Product – 题解

主要思路

硬推就完事了:

\[ \begin{aligned} g(n) &= \sum_{i = 1}^n \frac{n}{\gcd(i, n)} = n \sum_{d|n} \frac{1}{d} \sum_{i = 1}^n [\gcd(n, i) = d] \\ &= n \sum_{d|n} \frac{1}{d} \sum_{i = 1}^{\frac{n}{d}} [\gcd(\frac{n}{d}, i) = 1] = \sum_{d|n} \frac{n}{d} \varphi(\frac{n}{d}) = \sum_{d|n} d \varphi(d) \\ &= \sum_{d|n} d^2 \prod_{p_i} (1 – \frac{1}{p_i}) = \prod_{p_i} (1 + \sum_{g = 1}^{e_i} p_i^{2g} (1 – \frac{1}{p_i}) ) \\ &= \prod_{p_i} (1 + \sum_{g = 1}^{e_i} p_i^{2g} – \sum_{g = 1}^{e_i} p_i^{2g – 1}) = \prod_{p_i} (1 + \sum_{g = 1}^{e_i} p_i^{2g} – \frac{ \sum_{g = 1}^{e_i} p_i^{2g} }{p_i} ) \\ &= \prod_{p_i} (1 + (1 – \frac{1}{p_i}) (\sum_{g = 1}^{e_i} p_i^{2g}) ) = \prod_{p_i} (1 + (1 – \frac{1}{p_i}) \frac{p_i^2 (1 – p_i^{2e_i})}{1 – p_i^2}) \\ &= \prod_{p_i} (1 + \frac{(p_i – 1)}{p_i} \frac{p_i^2 (1 – p_i^{2e_i})}{(1 – p_i)(1 + p_i)} ) \\ &= \prod_{p_i} (1 – \frac{p_i (1 – p_i^{2e_i})}{1 + p_i} ) = \prod_{p_i} \frac{1 + p_i – p_i + p_i^{2e_i + 1}}{1 + p_i} \\ &= \prod_{p_i} \frac{p_i^{2e_i + 1} + 1}{1 + p_i} \end{aligned} \]

最后上下除一下,得到:

\[ \frac{f(n)}{g(n)} = \prod_{p_i} (1 + p_i) \]

代码

// QOJ2225.cpp
#include <bits/stdc++.h>

using namespace std;

const int mod = 1e9 + 7;

typedef long long ll;

ll gcd(ll a, ll b) { return b == 0 ? a : gcd(b, a % b); }

const int base[] = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37};

ll multiply(ll x, ll y, ll cmod)
{
    ll tmp = (x * y - (ll)((long double)x / cmod * y + 1.0e-8) * cmod) % cmod;
    return tmp < 0 ? tmp + cmod : tmp;
}

ll fpow(ll bas, ll tim, ll cmod)
{
    ll ret = 1;
    while (tim)
    {
        if (tim & 1)
            ret = multiply(ret, bas, cmod);
        bas = multiply(bas, bas, cmod);
        tim >>= 1;
    }
    return ret;
}

bool millerRabin(ll curt)
{
    for (auto cbase : base)
    {
        if (cbase == curt)
            return true;
        else if (cbase > curt)
            return false;
        ll x = fpow(cbase, curt - 1, curt), idx = curt - 1;
        if (x != 1)
            return false;
        while (x == 1 && idx % 2 == 0)
        {
            idx /= 2, x = fpow(cbase, idx, curt);
            if (x != 1 && x != curt - 1)
                return false;
        }
    }
    return true;
}

ll pollardRho(ll n)
{
    if (n % 2 == 0)
        return 2;
    if (n % 3 == 0)
        return 3;
    ll x = 0, y = 0, c = rand() % (n - 1) + 1, bas = 1, unit = (1 << 7), factor = 1;
    for (ll step = 2;; step <<= 1, y = x, bas = 1)
    {
        for (ll i = 1; i <= step; i++)
        {
            x = (multiply(x, x, n) + c) % n;
            bas = multiply(bas, abs(x - y), n);
            if (i % unit == 0)
            {
                factor = gcd(bas, n);
                if (factor > 1)
                    return factor;
            }
        }
        if (factor > 1 || ((factor = gcd(bas, n)) > 1))
            return factor;
    }
    return n;
}

vector<ll> p;

void factorize(ll x)
{
    if (x == 1)
        return;
    if (millerRabin(x))
        return (void)(p.push_back(x));
    ll p = x;
    while (p == x)
        p = pollardRho(x);
    while (x % p == 0)
        x /= p;
    factorize(p), factorize(x);
}

int main()
{
    int T;
    scanf("%d", &T);
    while (T--)
    {
        ll x;
        scanf("%lld", &x), p.clear(), factorize(x);
        sort(p.begin(), p.end()), p.erase(unique(p.begin(), p.end()), p.end());
        int ans = 1;
        for (ll x : p)
            ans = 1LL * ans * (x % mod + 1) % mod;
        printf("%d\n", ans);
    }
    return 0;
}

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