結果

問題 No.886 Direct
ユーザー hitonanode
提出日時 2022-01-29 16:15:50
言語 C++23
(gcc 13.3.0 + boost 1.87.0)
結果
AC  
実行時間 181 ms / 4,000 ms
コード長 15,280 bytes
コンパイル時間 1,287 ms
コンパイル使用メモリ 112,512 KB
実行使用メモリ 26,632 KB
最終ジャッジ日時 2025-01-02 07:35:25
合計ジャッジ時間 4,222 ms
ジャッジサーバーID
（参考情報） 		judge1 / judge3
このコードへのチャレンジ
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ファイルパターン 結果
sample AC * 4
other AC * 32
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ソースコード

diff #
プレゼンテーションモードにする

#line 1 "number/test/multiple_moebius.yuki886.test.cpp"
#define PROBLEM "https://yukicoder.me/problems/no/886"
#line 2 "modint.hpp"
#include <iostream>
#include <set>
#include <vector>
// CUT begin
template <int md> struct ModInt {
#if __cplusplus >= 201402L
#define MDCONST constexpr
#else
#define MDCONST
#endif
    using lint = long long;
    MDCONST static int mod() { return md; }
    static int get_primitive_root() {
        static int primitive_root = 0;
        if (!primitive_root) {
            primitive_root = [&]() {
                std::set<int> fac;
                int v = md - 1;
                for (lint i = 2; i * i <= v; i++)
                    while (v % i == 0) fac.insert(i), v /= i;
                if (v > 1) fac.insert(v);
                for (int g = 1; g < md; g++) {
                    bool ok = true;
                    for (auto i : fac)
                        if (ModInt(g).pow((md - 1) / i) == 1) {
                            ok = false;
                            break;
                        }
                    if (ok) return g;
                }
                return -1;
            }();
        }
        return primitive_root;
    }
    int val;
    MDCONST ModInt() : val(0) {}
    MDCONST ModInt &_setval(lint v) { return val = (v >= md ? v - md : v), *this; }
    MDCONST ModInt(lint v) { _setval(v % md + md); }
    MDCONST explicit operator bool() const { return val != 0; }
    MDCONST ModInt operator+(const ModInt &x) const { return ModInt()._setval((lint)val + x.val); }
    MDCONST ModInt operator-(const ModInt &x) const {
        return ModInt()._setval((lint)val - x.val + md);
    }
    MDCONST ModInt operator*(const ModInt &x) const {
        return ModInt()._setval((lint)val * x.val % md);
    }
    MDCONST ModInt operator/(const ModInt &x) const {
        return ModInt()._setval((lint)val * x.inv() % md);
    }
    MDCONST ModInt operator-() const { return ModInt()._setval(md - val); }
    MDCONST ModInt &operator+=(const ModInt &x) { return *this = *this + x; }
    MDCONST ModInt &operator-=(const ModInt &x) { return *this = *this - x; }
    MDCONST ModInt &operator*=(const ModInt &x) { return *this = *this * x; }
    MDCONST ModInt &operator/=(const ModInt &x) { return *this = *this / x; }
    friend MDCONST ModInt operator+(lint a, const ModInt &x) {
        return ModInt()._setval(a % md + x.val);
    }
    friend MDCONST ModInt operator-(lint a, const ModInt &x) {
        return ModInt()._setval(a % md - x.val + md);
    }
    friend MDCONST ModInt operator*(lint a, const ModInt &x) {
        return ModInt()._setval(a % md * x.val % md);
    }
    friend MDCONST ModInt operator/(lint a, const ModInt &x) {
        return ModInt()._setval(a % md * x.inv() % md);
    }
    MDCONST bool operator==(const ModInt &x) const { return val == x.val; }
    MDCONST bool operator!=(const ModInt &x) const { return val != x.val; }
    MDCONST bool operator<(const ModInt &x) const {
        return val < x.val;
    } // To use std::map<ModInt, T>
    friend std::istream &operator>>(std::istream &is, ModInt &x) {
        lint t;
        return is >> t, x = ModInt(t), is;
    }
    MDCONST friend std::ostream &operator<<(std::ostream &os, const ModInt &x) {
        return os << x.val;
    }
    MDCONST ModInt pow(lint n) const {
        ModInt ans = 1, tmp = *this;
        while (n) {
            if (n & 1) ans *= tmp;
            tmp *= tmp, n >>= 1;
        }
        return ans;
    }
    static std::vector<ModInt> facs, facinvs, invs;
    MDCONST static void _precalculation(int N) {
        int l0 = facs.size();
        if (N > md) N = md;
        if (N <= l0) return;
        facs.resize(N), facinvs.resize(N), invs.resize(N);
        for (int i = l0; i < N; i++) facs[i] = facs[i - 1] * i;
        facinvs[N - 1] = facs.back().pow(md - 2);
        for (int i = N - 2; i >= l0; i--) facinvs[i] = facinvs[i + 1] * (i + 1);
        for (int i = N - 1; i >= l0; i--) invs[i] = facinvs[i] * facs[i - 1];
    }
    MDCONST lint inv() const {
        if (this->val < std::min(md >> 1, 1 << 21)) {
            while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
            return invs[this->val].val;
        } else {
            return this->pow(md - 2).val;
        }
    }
    MDCONST ModInt fac() const {
        while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
        return facs[this->val];
    }
    MDCONST ModInt facinv() const {
        while (this->val >= int(facs.size())) _precalculation(facs.size() * 2);
        return facinvs[this->val];
    }
    MDCONST ModInt doublefac() const {
        lint k = (this->val + 1) / 2;
        return (this->val & 1) ? ModInt(k * 2).fac() / (ModInt(2).pow(k) * ModInt(k).fac())
                               : ModInt(k).fac() * ModInt(2).pow(k);
    }
    MDCONST ModInt nCr(const ModInt &r) const {
        return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv() * r.facinv();
    }
    MDCONST ModInt nPr(const ModInt &r) const {
        return (this->val < r.val) ? 0 : this->fac() * (*this - r).facinv();
    }
    ModInt sqrt() const {
        if (val == 0) return 0;
        if (md == 2) return val;
        if (pow((md - 1) / 2) != 1) return 0;
        ModInt b = 1;
        while (b.pow((md - 1) / 2) == 1) b += 1;
        int e = 0, m = md - 1;
        while (m % 2 == 0) m >>= 1, e++;
        ModInt x = pow((m - 1) / 2), y = (*this) * x * x;
        x *= (*this);
        ModInt z = b.pow(m);
        while (y != 1) {
            int j = 0;
            ModInt t = y;
            while (t != 1) j++, t *= t;
            z = z.pow(1LL << (e - j - 1));
            x *= z, z *= z, y *= z;
            e = j;
        }
        return ModInt(std::min(x.val, md - x.val));
    }
};
template <int md> std::vector<ModInt<md>> ModInt<md>::facs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::facinvs = {1};
template <int md> std::vector<ModInt<md>> ModInt<md>::invs = {0};
// using mint = ModInt<998244353>;
// using mint = ModInt<1000000007>;
#line 2 "number/sieve.hpp"
#include <cassert>
#include <map>
#line 5 "number/sieve.hpp"
// CUT begin
// Linear sieve algorithm for fast prime factorization
// Complexity: O(N) time, O(N) space:
// - MAXN = 10^7:  ~44 MB,  80~100 ms (Codeforces / AtCoder GCC, C++17)
// - MAXN = 10^8: ~435 MB, 810~980 ms (Codeforces / AtCoder GCC, C++17)
// Reference:
// [1] D. Gries, J. Misra, "A Linear Sieve Algorithm for Finding Prime Numbers,"
//     Communications of the ACM, 21(12), 999-1003, 1978.
// - https://cp-algorithms.com/algebra/prime-sieve-linear.html
// - https://37zigen.com/linear-sieve/
struct Sieve {
    std::vector<int> min_factor;
    std::vector<int> primes;
    Sieve(int MAXN) : min_factor(MAXN + 1) {
        for (int d = 2; d <= MAXN; d++) {
            if (!min_factor[d]) {
                min_factor[d] = d;
                primes.emplace_back(d);
            }
            for (const auto &p : primes) {
                if (p > min_factor[d] or d * p > MAXN) break;
                min_factor[d * p] = p;
            }
        }
    }
    // Prime factorization for 1 <= x <= MAXN^2
    // Complexity: O(log x)           (x <= MAXN)
    //             O(MAXN / log MAXN) (MAXN < x <= MAXN^2)
    template <class T> std::map<T, int> factorize(T x) const {
        std::map<T, int> ret;
        assert(x > 0 and
               x <= ((long long)min_factor.size() - 1) * ((long long)min_factor.size() - 1));
        for (const auto &p : primes) {
            if (x < T(min_factor.size())) break;
            while (!(x % p)) x /= p, ret[p]++;
        }
        if (x >= T(min_factor.size())) ret[x]++, x = 1;
        while (x > 1) ret[min_factor[x]]++, x /= min_factor[x];
        return ret;
    }
    // Enumerate divisors of 1 <= x <= MAXN^2
    // Be careful of highly composite numbers https://oeis.org/A002182/list
    // https://gist.github.com/dario2994/fb4713f252ca86c1254d#file-list-txt (n, (# of div. of n)):
    // 45360->100, 735134400(<1e9)->1344, 963761198400(<1e12)->6720
    template <class T> std::vector<T> divisors(T x) const {
        std::vector<T> ret{1};
        for (const auto p : factorize(x)) {
            int n = ret.size();
            for (int i = 0; i < n; i++) {
                for (T a = 1, d = 1; d <= p.second; d++) {
                    a *= p.first;
                    ret.push_back(ret[i] * a);
                }
            }
        }
        return ret; // NOT sorted
    }
    // Euler phi functions of divisors of given x
    // Verified: ABC212 G https://atcoder.jp/contests/abc212/tasks/abc212_g
    // Complexity: O(sqrt(x) + d(x))
    template <class T> std::map<T, T> euler_of_divisors(T x) const {
        assert(x >= 1);
        std::map<T, T> ret;
        ret[1] = 1;
        std::vector<T> divs{1};
        for (auto p : factorize(x)) {
            int n = ret.size();
            for (int i = 0; i < n; i++) {
                ret[divs[i] * p.first] = ret[divs[i]] * (p.first - 1);
                divs.push_back(divs[i] * p.first);
                for (T a = divs[i] * p.first, d = 1; d < p.second; a *= p.first, d++) {
                    ret[a * p.first] = ret[a] * p.first;
                    divs.push_back(a * p.first);
                }
            }
        }
        return ret;
    }
    // Moebius function Table, (-1)^{# of different prime factors} for square-free x
    // return: [0=>0, 1=>1, 2=>-1, 3=>-1, 4=>0, 5=>-1, 6=>1, 7=>-1, 8=>0, ...] https://oeis.org/A008683
    std::vector<int> GenerateMoebiusFunctionTable() const {
        std::vector<int> ret(min_factor.size());
        for (unsigned i = 1; i < min_factor.size(); i++) {
            if (i == 1) {
                ret[i] = 1;
            } else if ((i / min_factor[i]) % min_factor[i] == 0) {
                ret[i] = 0;
            } else {
                ret[i] = -ret[i / min_factor[i]];
            }
        }
        return ret;
    }
    // Calculate [0^K, 1^K, ..., nmax^K] in O(nmax)
    // Note: **0^0 == 1**
    template <class MODINT> std::vector<MODINT> enumerate_kth_pows(long long K, int nmax) const {
        assert(nmax < int(min_factor.size()));
        assert(K >= 0);
        if (K == 0) return std::vector<MODINT>(nmax + 1, 1);
        std::vector<MODINT> ret(nmax + 1);
        ret[0] = 0, ret[1] = 1;
        for (int n = 2; n <= nmax; n++) {
            if (min_factor[n] == n) {
                ret[n] = MODINT(n).pow(K);
            } else {
                ret[n] = ret[n / min_factor[n]] * ret[min_factor[n]];
            }
        }
        return ret;
    }
};
// Sieve sieve((1 << 20));
#line 3 "number/zeta_moebius_transform.hpp"
#include <algorithm>
#line 5 "number/zeta_moebius_transform.hpp"
#include <utility>
#line 7 "number/zeta_moebius_transform.hpp"
// f[n] に対して、全ての n の倍数 n*i に対する f[n*i] の和が出てくる 計算量 O(N loglog N)
// 素数p毎に処理する高速ゼータ変換
// 使用例 https://yukicoder.me/submissions/385043
template <class T> void multiple_zeta(std::vector<T> &f) {
    int N = int(f.size()) - 1;
    std::vector<int> is_prime(N + 1, 1);
    for (int p = 2; p <= N; p++) {
        if (is_prime[p]) {
            for (int q = p * 2; q <= N; q += p) is_prime[q] = 0;
            for (int j = N / p; j > 0; --j) f[j] += f[j * p];
        }
    }
}
// inverse of multiple_zeta O(N loglog N)
// 使用例 https://yukicoder.me/submissions/385120
template <class T> void multiple_moebius(std::vector<T> &f) {
    int N = int(f.size()) - 1;
    std::vector<int> is_prime(N + 1, 1);
    for (int p = 2; p <= N; p++) {
        if (is_prime[p]) {
            for (int q = p * 2; q <= N; q += p) is_prime[q] = 0;
            for (int j = 1; j * p <= N; ++j) f[j] -= f[j * p];
        }
    }
}
// f[n] に関して、全ての n の約数 m に対する f[m] の総和が出てくる 計算量 O(N loglog N)
template <class T> void divisor_zeta(std::vector<T> &f) {
    int N = int(f.size()) - 1;
    std::vector<int> is_prime(N + 1, 1);
    for (int p = 2; p <= N; ++p) {
        if (is_prime[p]) {
            for (int q = p * 2; q <= N; q += p) is_prime[q] = 0;
            for (int j = 1; j * p <= N; ++j) f[j * p] += f[j];
        }
    }
}
// inverse of divisor_zeta()
template <class T> void divisor_moebius(std::vector<T> &f) {
    int N = int(f.size()) - 1;
    std::vector<int> is_prime(N + 1, 1);
    for (int p = 2; p <= N; ++p) {
        if (is_prime[p]) {
            for (int q = p * 2; q <= N; q += p) is_prime[q] = 0;
            for (int j = N / p; j > 0; --j) f[j * p] -= f[j];
        }
    }
}
// GCD convolution
// ret[k] = \sum_{gcd(i, j) = k} f[i] * g[j]
template <class T> std::vector<T> gcdconv(std::vector<T> f, std::vector<T> g) {
    assert(f.size() == g.size());
    multiple_zeta(f);
    multiple_zeta(g);
    for (int i = 0; i < int(g.size()); ++i) f[i] *= g[i];
    multiple_moebius(f);
    return f;
}
// LCM convolution
// ret[k] = \sum_{lcm(i, j) = k} f[i] * g[j]
template <class T> std::vector<T> lcmconv(std::vector<T> f, std::vector<T> g) {
    assert(f.size() == g.size());
    divisor_zeta(f);
    divisor_zeta(g);
    for (int i = 0; i < int(g.size()); ++i) f[i] *= g[i];
    divisor_moebius(f);
    return f;
}
// fast_integer_moebius の高速化（登場しない素因数が多ければ計算量改善）
// Requirement: f の key として登場する正整数の全ての約数が key として登場
// Verified: https://toph.co/p/height-of-the-trees
template <typename Int, typename Val>
void sparse_fast_integer_moebius(std::vector<std::pair<Int, Val>> &f, const Sieve &sieve) {
    if (f.empty()) return;
    std::sort(f.begin(), f.end());
    assert(f.back().first < Int(sieve.min_factor.size()));
    std::vector<Int> primes;
    for (auto p : f) {
        if (sieve.min_factor[p.first] == p.first) primes.push_back(p.first);
    }
    std::vector<std::vector<int>> p2is(primes.size());
    for (int i = 0; i < int(f.size()); i++) {
        Int a = f[i].first, pold = 0;
        int k = 0;
        while (a > 1) {
            auto p = sieve.min_factor[a];
            if (p != pold) {
                while (primes[k] != p) k++;
                p2is[k].emplace_back(i);
            }
            pold = p, a /= p;
        }
    }
    for (int d = 0; d < int(primes.size()); d++) {
        Int p = primes[d];
        for (auto i : p2is[d]) {
            auto comp = [](const std::pair<Int, Val> &l, const std::pair<Int, Val> &r) {
                return l.first < r.first;
            };
            auto itr = std::lower_bound(f.begin(), f.end(), std::make_pair(f[i].first / p, 0), comp);
            itr->second -= f[i].second;
        }
    }
}
#line 7 "number/test/multiple_moebius.yuki886.test.cpp"
using namespace std;
using mint = ModInt<1000000007>;
int main() {
    int H, W;
    cin >> H >> W;
    int N = max(H, W);
    vector<mint> dp(N);
    auto calc = [&](int n, int W) -> mint {
        // 2 * ((W - n) + (W - 2 * n) + ...)
        mint k = W / n;
        return W * k * 2 - k * (k + 1) * n;
    };
    for (int n = 1; n < N; ++n) {
        mint n1 = calc(n, H), n2 = calc(n, W);
        dp[n] = (n1 + H) * (n2 + W) - mint(H) * W;
    }
    multiple_moebius(dp);
    cout << dp[1] / 2 << '\n';
}
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