test/yosupo/sum_of_totient_function.test.cpp

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Last update: 2024-01-08 17:31:43+09:00
Problem: https://judge.yosupo.jp/problem/sum_of_totient_function

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Code

#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_totient_function"

#include <bits/stdc++.h>

#include "../../math/modint.hpp"
#include "../../math/number-theory/euler_totient.hpp"
using namespace std;

using mint = Modint<998244353>;

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    long long N;
    cin >> N;
    auto [small, large] = totient_summatory_table<mint>(N);
    mint ans = large[1];
    cout << ans << endl;
}

#line 1 "test/yosupo/sum_of_totient_function.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/sum_of_totient_function"

#include <bits/stdc++.h>

#line 4 "math/modint.hpp"

/**
 * @brief Mod int
 */
template <int m>
class Modint {
    using mint = Modint;
    static_assert(m > 0, "Modulus must be positive");

   public:
    static constexpr int mod() { return m; }

    constexpr Modint(long long y = 0) : x(y >= 0 ? y % m : (y % m + m) % m) {}

    constexpr int val() const { return x; }

    constexpr mint& operator+=(const mint& r) {
        if ((x += r.x) >= m) x -= m;
        return *this;
    }
    constexpr mint& operator-=(const mint& r) {
        if ((x += m - r.x) >= m) x -= m;
        return *this;
    }
    constexpr mint& operator*=(const mint& r) {
        x = static_cast<int>(1LL * x * r.x % m);
        return *this;
    }
    constexpr mint& operator/=(const mint& r) { return *this *= r.inv(); }

    constexpr bool operator==(const mint& r) const { return x == r.x; }

    constexpr mint operator+() const { return *this; }
    constexpr mint operator-() const { return mint(-x); }

    constexpr friend mint operator+(const mint& l, const mint& r) {
        return mint(l) += r;
    }
    constexpr friend mint operator-(const mint& l, const mint& r) {
        return mint(l) -= r;
    }
    constexpr friend mint operator*(const mint& l, const mint& r) {
        return mint(l) *= r;
    }
    constexpr friend mint operator/(const mint& l, const mint& r) {
        return mint(l) /= r;
    }

    constexpr mint inv() const {
        int a = x, b = m, u = 1, v = 0;
        while (b > 0) {
            int t = a / b;
            std::swap(a -= t * b, b);
            std::swap(u -= t * v, v);
        }
        return mint(u);
    }

    constexpr mint pow(long long n) const {
        mint ret(1), mul(x);
        while (n > 0) {
            if (n & 1) ret *= mul;
            mul *= mul;
            n >>= 1;
        }
        return ret;
    }

    friend std::ostream& operator<<(std::ostream& os, const mint& r) {
        return os << r.x;
    }

    friend std::istream& operator>>(std::istream& is, mint& r) {
        long long t;
        is >> t;
        r = mint(t);
        return is;
    }

   private:
    int x;
};
#line 4 "math/number-theory/euler_totient.hpp"

long long euler_totient(long long n) {
    long long ret = n;
    if (n % 2 == 0) {
        ret -= ret / 2;
        while (n % 2 == 0) n /= 2;
    }
    for (long long i = 3; i * i <= n; i += 2) {
        if (n % i == 0) {
            ret -= ret / i;
            while (n % i == 0) n /= i;
        }
    }
    if (n != 1) ret -= ret / n;
    return ret;
}

std::vector<int> euler_totient_table(int n) {
    std::vector<int> ret(n + 1);
    std::iota(ret.begin(), ret.end(), 0);
    for (int i = 2; i <= n; ++i) {
        if (ret[i] == i) {
            for (int j = i; j <= n; j += i) {
                ret[j] = ret[j] / i * (i - 1);
            }
        }
    }
    return ret;
}

template <typename mint>
std::pair<std::vector<mint>, std::vector<mint>> totient_summatory_table(
    long long n) {
    if (n == 0) return {{0}, {0}};
    const int b = std::min(n, (long long)1e4);
    std::vector<mint> small(n / b + 1), large(b + 1);

    std::vector<int> totient(n / b + 1);
    std::iota(totient.begin(), totient.end(), 0);
    for (int i = 2; i <= n / b; ++i) {
        if (totient[i] != i) continue;
        for (int j = i; j <= n / b; j += i) {
            totient[j] = totient[j] / i * (i - 1);
        }
    }
    for (int i = 0; i < n / b; ++i) small[i + 1] = small[i] + totient[i + 1];

    for (int i = 1; i <= b; ++i) {
        mint k = n / i;
        large[i] = k * (k + 1) / 2;
    }
    for (long long i = b; i >= 1; --i) {
        for (long long l = 2; l <= n / i;) {
            long long q = n / (i * l), r = n / (i * q) + 1;
            large[i] -=
                (i * l <= b ? large[i * l] : small[n / (i * l)]) * (r - l);
            l = r;
        }
    }
    return {small, large};
}
#line 7 "test/yosupo/sum_of_totient_function.test.cpp"
using namespace std;

using mint = Modint<998244353>;

int main() {
    ios_base::sync_with_stdio(false);
    cin.tie(nullptr);

    long long N;
    cin >> N;
    auto [small, large] = totient_summatory_table<mint>(N);
    mint ans = large[1];
    cout << ans << endl;
}