tests/fds_with_prefix_sums/sum_of_totient_function.test.cpp

• View this file on GitHub
• Last update: 2026-01-25 23:49:58+09:00
• Problem: https://judge.yosupo.jp/problem/sum_of_totient_function

Depends on

• Formal Dirichlet series with prefix sums (tools/fds_with_prefix_sums.hpp)
• $\left\lfloor \sqrt{x} \right\rfloor$ (tools/floor_sqrt.hpp)
• Concept for a type compatible with atcoder::static_modint (tools/modint_compatible.hpp)

Code

// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/sum_of_totient_function

#include <iostream>
#include "atcoder/modint.hpp"
#include "tools/fds_with_prefix_sums.hpp"

using mint = atcoder::modint998244353;

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

  long long N;
  std::cin >> N;

  const auto inv2 = mint::raw(2).inv();
  std::cout << (
    tools::fds_with_prefix_sums<mint>(N, [&](const long long i) { return mint(i) * mint(i + 1) * inv2; }) /
    tools::fds_with_prefix_sums<mint>(N, [&](const long long i) { return mint(i); })
  ).rbegin()->val() << '\n';

  return 0;
}

#line 1 "tests/fds_with_prefix_sums/sum_of_totient_function.test.cpp"
// competitive-verifier: PROBLEM https://judge.yosupo.jp/problem/sum_of_totient_function

#include <iostream>
#line 1 "lib/ac-library/atcoder/modint.hpp"



#include <cassert>
#include <numeric>
#include <type_traits>

#ifdef _MSC_VER
#include <intrin.h>
#endif

#line 1 "lib/ac-library/atcoder/internal_math.hpp"



#include <utility>

#ifdef _MSC_VER
#include <intrin.h>
#endif

namespace atcoder {

namespace internal {

// @param m `1 <= m`
// @return x mod m
constexpr long long safe_mod(long long x, long long m) {
    x %= m;
    if (x < 0) x += m;
    return x;
}

// Fast modular multiplication by barrett reduction
// Reference: https://en.wikipedia.org/wiki/Barrett_reduction
// NOTE: reconsider after Ice Lake
struct barrett {
    unsigned int _m;
    unsigned long long im;

    // @param m `1 <= m`
    explicit barrett(unsigned int m) : _m(m), im((unsigned long long)(-1) / m + 1) {}

    // @return m
    unsigned int umod() const { return _m; }

    // @param a `0 <= a < m`
    // @param b `0 <= b < m`
    // @return `a * b % m`
    unsigned int mul(unsigned int a, unsigned int b) const {
        // [1] m = 1
        // a = b = im = 0, so okay

        // [2] m >= 2
        // im = ceil(2^64 / m)
        // -> im * m = 2^64 + r (0 <= r < m)
        // let z = a*b = c*m + d (0 <= c, d < m)
        // a*b * im = (c*m + d) * im = c*(im*m) + d*im = c*2^64 + c*r + d*im
        // c*r + d*im < m * m + m * im < m * m + 2^64 + m <= 2^64 + m * (m + 1) < 2^64 * 2
        // ((ab * im) >> 64) == c or c + 1
        unsigned long long z = a;
        z *= b;
#ifdef _MSC_VER
        unsigned long long x;
        _umul128(z, im, &x);
#else
        unsigned long long x =
            (unsigned long long)(((unsigned __int128)(z)*im) >> 64);
#endif
        unsigned long long y = x * _m;
        return (unsigned int)(z - y + (z < y ? _m : 0));
    }
};

// @param n `0 <= n`
// @param m `1 <= m`
// @return `(x ** n) % m`
constexpr long long pow_mod_constexpr(long long x, long long n, int m) {
    if (m == 1) return 0;
    unsigned int _m = (unsigned int)(m);
    unsigned long long r = 1;
    unsigned long long y = safe_mod(x, m);
    while (n) {
        if (n & 1) r = (r * y) % _m;
        y = (y * y) % _m;
        n >>= 1;
    }
    return r;
}

// Reference:
// M. Forisek and J. Jancina,
// Fast Primality Testing for Integers That Fit into a Machine Word
// @param n `0 <= n`
constexpr bool is_prime_constexpr(int n) {
    if (n <= 1) return false;
    if (n == 2 || n == 7 || n == 61) return true;
    if (n % 2 == 0) return false;
    long long d = n - 1;
    while (d % 2 == 0) d /= 2;
    constexpr long long bases[3] = {2, 7, 61};
    for (long long a : bases) {
        long long t = d;
        long long y = pow_mod_constexpr(a, t, n);
        while (t != n - 1 && y != 1 && y != n - 1) {
            y = y * y % n;
            t <<= 1;
        }
        if (y != n - 1 && t % 2 == 0) {
            return false;
        }
    }
    return true;
}
template <int n> constexpr bool is_prime = is_prime_constexpr(n);

// @param b `1 <= b`
// @return pair(g, x) s.t. g = gcd(a, b), xa = g (mod b), 0 <= x < b/g
constexpr std::pair<long long, long long> inv_gcd(long long a, long long b) {
    a = safe_mod(a, b);
    if (a == 0) return {b, 0};

    // Contracts:
    // [1] s - m0 * a = 0 (mod b)
    // [2] t - m1 * a = 0 (mod b)
    // [3] s * |m1| + t * |m0| <= b
    long long s = b, t = a;
    long long m0 = 0, m1 = 1;

    while (t) {
        long long u = s / t;
        s -= t * u;
        m0 -= m1 * u;  // |m1 * u| <= |m1| * s <= b

        // [3]:
        // (s - t * u) * |m1| + t * |m0 - m1 * u|
        // <= s * |m1| - t * u * |m1| + t * (|m0| + |m1| * u)
        // = s * |m1| + t * |m0| <= b

        auto tmp = s;
        s = t;
        t = tmp;
        tmp = m0;
        m0 = m1;
        m1 = tmp;
    }
    // by [3]: |m0| <= b/g
    // by g != b: |m0| < b/g
    if (m0 < 0) m0 += b / s;
    return {s, m0};
}

// Compile time primitive root
// @param m must be prime
// @return primitive root (and minimum in now)
constexpr int primitive_root_constexpr(int m) {
    if (m == 2) return 1;
    if (m == 167772161) return 3;
    if (m == 469762049) return 3;
    if (m == 754974721) return 11;
    if (m == 998244353) return 3;
    int divs[20] = {};
    divs[0] = 2;
    int cnt = 1;
    int x = (m - 1) / 2;
    while (x % 2 == 0) x /= 2;
    for (int i = 3; (long long)(i)*i <= x; i += 2) {
        if (x % i == 0) {
            divs[cnt++] = i;
            while (x % i == 0) {
                x /= i;
            }
        }
    }
    if (x > 1) {
        divs[cnt++] = x;
    }
    for (int g = 2;; g++) {
        bool ok = true;
        for (int i = 0; i < cnt; i++) {
            if (pow_mod_constexpr(g, (m - 1) / divs[i], m) == 1) {
                ok = false;
                break;
            }
        }
        if (ok) return g;
    }
}
template <int m> constexpr int primitive_root = primitive_root_constexpr(m);

// @param n `n < 2^32`
// @param m `1 <= m < 2^32`
// @return sum_{i=0}^{n-1} floor((ai + b) / m) (mod 2^64)
unsigned long long floor_sum_unsigned(unsigned long long n,
                                      unsigned long long m,
                                      unsigned long long a,
                                      unsigned long long b) {
    unsigned long long ans = 0;
    while (true) {
        if (a >= m) {
            ans += n * (n - 1) / 2 * (a / m);
            a %= m;
        }
        if (b >= m) {
            ans += n * (b / m);
            b %= m;
        }

        unsigned long long y_max = a * n + b;
        if (y_max < m) break;
        // y_max < m * (n + 1)
        // floor(y_max / m) <= n
        n = (unsigned long long)(y_max / m);
        b = (unsigned long long)(y_max % m);
        std::swap(m, a);
    }
    return ans;
}

}  // namespace internal

}  // namespace atcoder


#line 1 "lib/ac-library/atcoder/internal_type_traits.hpp"



#line 7 "lib/ac-library/atcoder/internal_type_traits.hpp"

namespace atcoder {

namespace internal {

#ifndef _MSC_VER
template <class T>
using is_signed_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value ||
                                  std::is_same<T, __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int128 =
    typename std::conditional<std::is_same<T, __uint128_t>::value ||
                                  std::is_same<T, unsigned __int128>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using make_unsigned_int128 =
    typename std::conditional<std::is_same<T, __int128_t>::value,
                              __uint128_t,
                              unsigned __int128>;

template <class T>
using is_integral = typename std::conditional<std::is_integral<T>::value ||
                                                  is_signed_int128<T>::value ||
                                                  is_unsigned_int128<T>::value,
                                              std::true_type,
                                              std::false_type>::type;

template <class T>
using is_signed_int = typename std::conditional<(is_integral<T>::value &&
                                                 std::is_signed<T>::value) ||
                                                    is_signed_int128<T>::value,
                                                std::true_type,
                                                std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<(is_integral<T>::value &&
                               std::is_unsigned<T>::value) ||
                                  is_unsigned_int128<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<
    is_signed_int128<T>::value,
    make_unsigned_int128<T>,
    typename std::conditional<std::is_signed<T>::value,
                              std::make_unsigned<T>,
                              std::common_type<T>>::type>::type;

#else

template <class T> using is_integral = typename std::is_integral<T>;

template <class T>
using is_signed_int =
    typename std::conditional<is_integral<T>::value && std::is_signed<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using is_unsigned_int =
    typename std::conditional<is_integral<T>::value &&
                                  std::is_unsigned<T>::value,
                              std::true_type,
                              std::false_type>::type;

template <class T>
using to_unsigned = typename std::conditional<is_signed_int<T>::value,
                                              std::make_unsigned<T>,
                                              std::common_type<T>>::type;

#endif

template <class T>
using is_signed_int_t = std::enable_if_t<is_signed_int<T>::value>;

template <class T>
using is_unsigned_int_t = std::enable_if_t<is_unsigned_int<T>::value>;

template <class T> using to_unsigned_t = typename to_unsigned<T>::type;

}  // namespace internal

}  // namespace atcoder


#line 14 "lib/ac-library/atcoder/modint.hpp"

namespace atcoder {

namespace internal {

struct modint_base {};
struct static_modint_base : modint_base {};

template <class T> using is_modint = std::is_base_of<modint_base, T>;
template <class T> using is_modint_t = std::enable_if_t<is_modint<T>::value>;

}  // namespace internal

template <int m, std::enable_if_t<(1 <= m)>* = nullptr>
struct static_modint : internal::static_modint_base {
    using mint = static_modint;

  public:
    static constexpr int mod() { return m; }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    static_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    static_modint(T v) {
        long long x = (long long)(v % (long long)(umod()));
        if (x < 0) x += umod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    static_modint(T v) {
        _v = (unsigned int)(v % umod());
    }

    int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v -= rhs._v;
        if (_v >= umod()) _v += umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        unsigned long long z = _v;
        z *= rhs._v;
        _v = (unsigned int)(z % umod());
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

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

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        if (prime) {
            assert(_v);
            return pow(umod() - 2);
        } else {
            auto eg = internal::inv_gcd(_v, m);
            assert(eg.first == 1);
            return eg.second;
        }
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static constexpr unsigned int umod() { return m; }
    static constexpr bool prime = internal::is_prime<m>;
};

template <int id> struct dynamic_modint : internal::modint_base {
    using mint = dynamic_modint;

  public:
    static int mod() { return (int)(bt.umod()); }
    static void set_mod(int m) {
        assert(1 <= m);
        bt = internal::barrett(m);
    }
    static mint raw(int v) {
        mint x;
        x._v = v;
        return x;
    }

    dynamic_modint() : _v(0) {}
    template <class T, internal::is_signed_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        long long x = (long long)(v % (long long)(mod()));
        if (x < 0) x += mod();
        _v = (unsigned int)(x);
    }
    template <class T, internal::is_unsigned_int_t<T>* = nullptr>
    dynamic_modint(T v) {
        _v = (unsigned int)(v % mod());
    }

    int val() const { return _v; }

    mint& operator++() {
        _v++;
        if (_v == umod()) _v = 0;
        return *this;
    }
    mint& operator--() {
        if (_v == 0) _v = umod();
        _v--;
        return *this;
    }
    mint operator++(int) {
        mint result = *this;
        ++*this;
        return result;
    }
    mint operator--(int) {
        mint result = *this;
        --*this;
        return result;
    }

    mint& operator+=(const mint& rhs) {
        _v += rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator-=(const mint& rhs) {
        _v += mod() - rhs._v;
        if (_v >= umod()) _v -= umod();
        return *this;
    }
    mint& operator*=(const mint& rhs) {
        _v = bt.mul(_v, rhs._v);
        return *this;
    }
    mint& operator/=(const mint& rhs) { return *this = *this * rhs.inv(); }

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

    mint pow(long long n) const {
        assert(0 <= n);
        mint x = *this, r = 1;
        while (n) {
            if (n & 1) r *= x;
            x *= x;
            n >>= 1;
        }
        return r;
    }
    mint inv() const {
        auto eg = internal::inv_gcd(_v, mod());
        assert(eg.first == 1);
        return eg.second;
    }

    friend mint operator+(const mint& lhs, const mint& rhs) {
        return mint(lhs) += rhs;
    }
    friend mint operator-(const mint& lhs, const mint& rhs) {
        return mint(lhs) -= rhs;
    }
    friend mint operator*(const mint& lhs, const mint& rhs) {
        return mint(lhs) *= rhs;
    }
    friend mint operator/(const mint& lhs, const mint& rhs) {
        return mint(lhs) /= rhs;
    }
    friend bool operator==(const mint& lhs, const mint& rhs) {
        return lhs._v == rhs._v;
    }
    friend bool operator!=(const mint& lhs, const mint& rhs) {
        return lhs._v != rhs._v;
    }

  private:
    unsigned int _v;
    static internal::barrett bt;
    static unsigned int umod() { return bt.umod(); }
};
template <int id> internal::barrett dynamic_modint<id>::bt(998244353);

using modint998244353 = static_modint<998244353>;
using modint1000000007 = static_modint<1000000007>;
using modint = dynamic_modint<-1>;

namespace internal {

template <class T>
using is_static_modint = std::is_base_of<internal::static_modint_base, T>;

template <class T>
using is_static_modint_t = std::enable_if_t<is_static_modint<T>::value>;

template <class> struct is_dynamic_modint : public std::false_type {};
template <int id>
struct is_dynamic_modint<dynamic_modint<id>> : public std::true_type {};

template <class T>
using is_dynamic_modint_t = std::enable_if_t<is_dynamic_modint<T>::value>;

}  // namespace internal

}  // namespace atcoder


#line 1 "tools/fds_with_prefix_sums.hpp"



#include <algorithm>
#line 6 "tools/fds_with_prefix_sums.hpp"
#include <concepts>
#include <cstddef>
#include <iterator>
#line 10 "tools/fds_with_prefix_sums.hpp"
#include <ranges>
#line 13 "tools/fds_with_prefix_sums.hpp"
#include <vector>
#line 1 "tools/floor_sqrt.hpp"



#line 5 "tools/floor_sqrt.hpp"

namespace tools {

  template <typename T>
  T floor_sqrt(const T n) {
    assert(n >= 0);

    T ok = 0;
    T ng;
    for (ng = 1; ng <= n / ng; ng *= 2);

    while (ng - ok > 1) {
      const T mid = ok + (ng - ok) / 2;
      if (mid <= n / mid) {
        ok = mid;
      } else {
        ng = mid;
      }
    }

    return ok;
  }
}


#line 1 "tools/modint_compatible.hpp"



#line 6 "tools/modint_compatible.hpp"

namespace tools {
  template <typename T>
  concept modint_compatible = std::regular<std::remove_cv_t<T>>
    && std::equality_comparable<std::remove_cv_t<T>>
    && std::constructible_from<std::remove_cv_t<T>, bool>
    && std::constructible_from<std::remove_cv_t<T>, char>
    && std::constructible_from<std::remove_cv_t<T>, int>
    && std::constructible_from<std::remove_cv_t<T>, long long>
    && std::constructible_from<std::remove_cv_t<T>, unsigned int>
    && std::constructible_from<std::remove_cv_t<T>, unsigned long long>
    && requires(std::remove_cv_t<T> a, std::remove_cv_t<T> b, int v_int, long long v_ll) {
      { std::remove_cv_t<T>::mod() } -> std::convertible_to<int>;
      { std::remove_cv_t<T>::raw(v_int) } -> std::same_as<std::remove_cv_t<T>>;
      { a.val() } -> std::convertible_to<int>;
      { ++a } -> std::same_as<std::remove_cv_t<T>&>;
      { --a } -> std::same_as<std::remove_cv_t<T>&>;
      { a++ } -> std::same_as<std::remove_cv_t<T>>;
      { a-- } -> std::same_as<std::remove_cv_t<T>>;
      { a += b } -> std::same_as<std::remove_cv_t<T>&>;
      { a -= b } -> std::same_as<std::remove_cv_t<T>&>;
      { a *= b } -> std::same_as<std::remove_cv_t<T>&>;
      { a /= b } -> std::same_as<std::remove_cv_t<T>&>;
      { +a } -> std::same_as<std::remove_cv_t<T>>;
      { -a } -> std::same_as<std::remove_cv_t<T>>;
      { a.pow(v_ll) } -> std::same_as<std::remove_cv_t<T>>;
      { a.inv() } -> std::same_as<std::remove_cv_t<T>>;
      { a + b } -> std::same_as<std::remove_cv_t<T>>;
      { a - b } -> std::same_as<std::remove_cv_t<T>>;
      { a * b } -> std::same_as<std::remove_cv_t<T>>;
      { a / b } -> std::same_as<std::remove_cv_t<T>>;
    };
}


#line 16 "tools/fds_with_prefix_sums.hpp"

namespace tools {
  template <tools::modint_compatible M>
  class fds_with_prefix_sums {
    using F = tools::fds_with_prefix_sums<M>;
    long long m_N;
    std::vector<M> m_lo;
    std::vector<M> m_hi;

    long long sqrt_N() const {
      return this->m_lo.size() - 1;
    }
    long long hi_max() const {
      return this->m_hi.size() - 1;
    }

  public:
    class iterator {
      const F *m_parent;
      std::size_t m_i;

    public:
      using reference = const M&;
      using value_type = M;
      using difference_type = std::ptrdiff_t;
      using pointer = const M*;
      using iterator_category = std::random_access_iterator_tag;

      iterator() = default;
      iterator(const F * const parent, const std::size_t i) : m_parent(parent), m_i(i) {
      }

      reference operator*() const {
        if (this->m_i + 1 < this->m_parent->m_lo.size()) {
          return this->m_parent->m_lo[this->m_i + 1];
        } else {
          return this->m_parent->m_hi[this->m_parent->m_lo.size() + this->m_parent->m_hi.size() - this->m_i - 2];
        }
      }
      pointer operator->() const {
        return &(*(*this));
      }

      iterator& operator++() {
        ++this->m_i;
        return *this;
      }
      iterator operator++(int) {
        const auto self = *this;
        ++*this;
        return self;
      }
      iterator& operator--() {
        --this->m_i;
        return *this;
      }
      iterator operator--(int) {
        const auto self = *this;
        --*this;
        return self;
      }
      iterator& operator+=(const difference_type n) {
        this->m_i += n;
        return *this;
      }
      iterator& operator-=(const difference_type n) {
        this->m_i -= n;
        return *this;
      }
      friend iterator operator+(const iterator self, const difference_type n) {
        return iterator(self.m_parent, self.m_i + n);
      }
      friend iterator operator+(const difference_type n, const iterator self) {
        return self + n;
      }
      friend iterator operator-(const iterator self, const difference_type n) {
        return iterator(self.m_parent, self.m_i - n);
      }
      friend difference_type operator-(const iterator lhs, const iterator rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return static_cast<difference_type>(lhs.m_i) - static_cast<difference_type>(rhs.m_i);
      }
      reference operator[](const difference_type n) const {
        return *(*this + n);
      }

      friend bool operator==(const iterator lhs, const iterator rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i == rhs.m_i;
      }
      friend bool operator!=(const iterator lhs, const iterator rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i != rhs.m_i;
      }
      friend bool operator<(const iterator lhs, const iterator rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i < rhs.m_i;
      }
      friend bool operator<=(const iterator lhs, const iterator rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i <= rhs.m_i;
      }
      friend bool operator>(const iterator lhs, const iterator rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i > rhs.m_i;
      }
      friend bool operator>=(const iterator lhs, const iterator rhs) {
        assert(lhs.m_parent == rhs.m_parent);
        return lhs.m_i >= rhs.m_i;
      }
    };
    using reverse_iterator = std::reverse_iterator<iterator>;

    fds_with_prefix_sums() = default;
    explicit fds_with_prefix_sums(const long long N) : fds_with_prefix_sums(N, [](long long) { return M::raw(0); }) {
    }
    template <typename PrefixSumFunction>
    requires std::regular_invocable<PrefixSumFunction, long long>
          && std::assignable_from<M&, std::invoke_result_t<PrefixSumFunction, long long>>
    fds_with_prefix_sums(const long long N, const PrefixSumFunction& sum) : m_N(N) {
      assert(N >= 1);
      const auto sqrt_N = tools::floor_sqrt(N);
      this->m_lo.reserve(sqrt_N + 1);
      this->m_lo.push_back(M::raw(0));
      for (long long i = 1; i <= sqrt_N; ++i) {
        this->m_lo.push_back(sum(i));
      }
      const auto hi_max = sqrt_N - (N < sqrt_N * (sqrt_N + 1));
      this->m_hi.reserve(hi_max + 1);
      this->m_hi.push_back(M::raw(0));
      for (long long i = 1; i <= hi_max; ++i) {
        this->m_hi.push_back(sum(N / i));
      }
    }
    template <std::ranges::input_range R>
    requires std::assignable_from<M&, std::ranges::range_value_t<R>>
    fds_with_prefix_sums(const long long N, R&& v) : m_N(N) {
      assert(N >= 1);
      const auto sqrt_N = tools::floor_sqrt(N);
      this->m_lo.resize(sqrt_N + 1);
      this->m_lo[0] = M::raw(0);
      const auto hi_max = sqrt_N - (N < sqrt_N * (sqrt_N + 1));
      this->m_hi.resize(hi_max + 1);
      this->m_hi[0] = M::raw(0);
      std::ranges::copy_n(std::ranges::copy_n(std::ranges::begin(v), sqrt_N, std::next(this->m_lo.begin())).in, hi_max, this->m_hi.rbegin());
    }

    iterator begin() const {
      return iterator(this, 0);
    }
    iterator end() const {
      return iterator(this, this->sqrt_N() + this->hi_max());
    }
    reverse_iterator rbegin() const {
      return std::make_reverse_iterator(this->end());
    }
    reverse_iterator rend() const {
      return std::make_reverse_iterator(this->begin());
    }

    F operator+() const {
      return *this;
    }
    F operator-() const {
      F g(this->m_N);
      for (int i = 1; i <= this->sqrt_N(); ++i) {
        g.m_lo[i] = -this->m_lo[i];
      }
      for (int i = 1; i <= this->hi_max(); ++i) {
        g.m_hi[i] = -this->m_hi[i];
      }
      return g;
    }

    F& operator+=(const F& g) {
      assert(this->m_N == g.m_N);
      for (int i = 1; i <= this->sqrt_N(); ++i) {
        this->m_lo[i] += g.m_lo[i];
      }
      for (int i = 1; i <= this->hi_max(); ++i) {
        this->m_hi[i] += g.m_hi[i];
      }
      return *this;
    }
    friend F operator+(const F& f, const F& g) {
      return F(f) += g;
    }

    F& operator-=(const F& g) {
      assert(this->m_N == g.m_N);
      for (int i = 1; i <= this->sqrt_N(); ++i) {
        this->m_lo[i] -= g.m_lo[i];
      }
      for (int i = 1; i <= this->hi_max(); ++i) {
        this->m_hi[i] -= g.m_hi[i];
      }
      return *this;
    }
    friend F operator-(const F& f, const F& g) {
      return F(f) -= g;
    }

    F& operator*=(const M& g) {
      return *this = *this * g;
    }
    friend F operator*(const F& f, const F& g) {
      assert(f.m_N == g.m_N);
      const auto hi_max = f.hi_max();

      F h(f.m_N);
      h.m_hi.resize(hi_max + 3);

      for (int i = 1; i <= h.sqrt_N(); ++i) {
        for (int j = 1; i * j <= h.sqrt_N(); ++j) {
          h.m_lo[i * j] += (f.m_lo[i] - f.m_lo[i - 1]) * (g.m_lo[j] - g.m_lo[j - 1]);
        }
      }
      for (int k = 1; k <= h.sqrt_N(); ++k) {
        h.m_lo[k] += h.m_lo[k - 1];
      }

      const auto apply = [&](const long long z_1, const long long z_2, const M d) {
        h.m_hi[z_1] += d;
        h.m_hi[z_2 + 1] -= d;
      };
      for (long long i = 1; i * (i + 1) * (i + 1) <= h.m_N; ++i) {
        for (long long j = i + 1; i * j * j <= h.m_N; ++j) {
          apply(j, std::min(h.m_N / (i * j), hi_max + 1), (f.m_lo[i] - f.m_lo[i - 1]) * (g.m_lo[j] - g.m_lo[j - 1]));
          apply(j, j, ((i * j <= hi_max ? f.m_hi[i * j] : f.m_lo[h.m_N / (i * j)]) - f.m_lo[j - 1]) * (g.m_lo[i] - g.m_lo[i - 1]));
          apply(i, i, (f.m_lo[j] - f.m_lo[j - 1]) * ((i * j <= hi_max ? g.m_hi[i * j] : g.m_lo[h.m_N / (i * j)]) - g.m_lo[j - 1]));
        }
      }
      for (long long i = 1; i * i * (i + 1) <= h.m_N; ++i) {
        for (long long j = i; i * j * (j + 1) <= h.m_N; ++j) {
          apply(i, i, ((i * j <= hi_max ? f.m_hi[i * j] : f.m_lo[h.m_N / (i * j)]) - f.m_lo[j]) * (g.m_lo[j] - g.m_lo[j - 1]));
          apply(j, j, (f.m_lo[i] - f.m_lo[i - 1]) * ((i * j <= hi_max ? g.m_hi[i * j] : g.m_lo[h.m_N / (i * j)]) - g.m_lo[j]));
          apply(j + 1, std::min(h.m_N / (i * j), hi_max + 1), (f.m_lo[j] - f.m_lo[j - 1]) * (g.m_lo[i] - g.m_lo[i - 1]));
        }
      }
      for (long long i = 1; i * i * i <= h.m_N; ++i) {
        apply(i, i, (f.m_lo[i] - f.m_lo[i - 1]) * (g.m_lo[i] - g.m_lo[i - 1]));
      }
      for (int i = 1; i <= hi_max; ++i) {
        h.m_hi[i] += h.m_hi[i - 1];
      }
      h.m_hi.erase(h.m_hi.end() - 2, h.m_hi.end());

      return h;
    }

    F& operator/=(const F& g) {
      assert(this->m_N == g.m_N);
      assert(std::gcd(g.m_lo[1].val(), M::mod()) == 1);
      const auto g1_inv = g.m_lo[1].inv();
      const auto hi_max = this->hi_max();

      for (int i = this->sqrt_N(); i >= 1; --i) {
        this->m_lo[i] -= this->m_lo[i - 1];
      }
      for (int i = hi_max; i >= 1; --i) {
        this->m_hi[i] -= this->m_hi[i - 1];
      }
      this->m_hi.resize(hi_max + 3);

      for (int i = 1; i <= this->sqrt_N(); ++i) {
        this->m_lo[i] *= g1_inv;
        for (int j = 2; i * j <= this->sqrt_N(); ++j) {
          this->m_lo[i * j] -= (g.m_lo[j] - g.m_lo[j - 1]) * this->m_lo[i];
        }
      }
      for (int i = 1; i <= this->sqrt_N(); ++i) {
        this->m_lo[i] += this->m_lo[i - 1];
      }

      const auto apply = [&](const long long z_1, const long long z_2, const M d) {
        this->m_hi[z_1] -= d;
        this->m_hi[z_2 + 1] += d;
      };
      for (long long i = 1; i * (i + 1) * (i + 1) <= this->m_N; ++i) {
        for (long long j = i + 1; i * j * j <= this->m_N; ++j) {
          apply(j, std::min(this->m_N / (i * j), hi_max + 1), (g.m_lo[i] - g.m_lo[i - 1]) * (this->m_lo[j] - this->m_lo[j - 1]));
          apply(j, j, ((i * j <= hi_max ? g.m_hi[i * j] : g.m_lo[this->m_N / (i * j)]) - g.m_lo[j - 1]) * (this->m_lo[i] - this->m_lo[i - 1]));
        }
      }
      for (long long i = 1; i * i * (i + 1) <= this->m_N; ++i) {
        for (long long j = i; i * j * (j + 1) <= this->m_N; ++j) {
          apply(i, i, ((i * j <= hi_max ? g.m_hi[i * j] : g.m_lo[this->m_N / (i * j)]) - g.m_lo[j]) * (this->m_lo[j] - this->m_lo[j - 1]));
          apply(j + 1, std::min(this->m_N / (i * j), hi_max + 1), (g.m_lo[j] - g.m_lo[j - 1]) * (this->m_lo[i] - this->m_lo[i - 1]));
        }
      }
      for (long long i = 1; i * i * i <= this->m_N; ++i) {
        apply(i, i, (g.m_lo[i] - g.m_lo[i - 1]) * (this->m_lo[i] - this->m_lo[i - 1]));
      }
      for (int i = 1; i <= hi_max; ++i) {
        this->m_hi[i] += this->m_hi[i - 1];
      }
      this->m_hi.erase(this->m_hi.end() - 2, this->m_hi.end());

      for (auto b = hi_max; b >= 1; --b) {
        for (auto j = b + 1; j * j <= this->m_N / b; ++j) {
          this->m_hi[b] -= (g.m_lo[j] - g.m_lo[j - 1]) * ((b * j <= hi_max ? this->m_hi[b * j] : this->m_lo[this->m_N / (b * j)]) - this->m_lo[j - 1]);
        }
        for (long long i = 2; i <= b && i * b * (b + 1) <= this->m_N; ++i) {
          this->m_hi[b] -= (g.m_lo[i] - g.m_lo[i - 1]) * ((i * b <= hi_max ? this->m_hi[i * b] : this->m_lo[this->m_N / (i * b)]) - this->m_lo[b]);
        }
        this->m_hi[b] += g.m_lo[1] * this->m_lo[b];
        this->m_hi[b] *= g1_inv;
      }

      return *this;
    }
    friend F operator/(const F& f, const F& g) {
      return F(f) /= g;
    }
  };
}


#line 6 "tests/fds_with_prefix_sums/sum_of_totient_function.test.cpp"

using mint = atcoder::modint998244353;

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

  long long N;
  std::cin >> N;

  const auto inv2 = mint::raw(2).inv();
  std::cout << (
    tools::fds_with_prefix_sums<mint>(N, [&](const long long i) { return mint(i) * mint(i + 1) * inv2; }) /
    tools::fds_with_prefix_sums<mint>(N, [&](const long long i) { return mint(i); })
  ).rbegin()->val() << '\n';

  return 0;
}

Test cases

Env	Name	Status	Elapsed	Memory
g++	boundaryA_00	AC	44 ms	4 MB
g++	boundaryA_01	AC	94 ms	5 MB
g++	boundaryA_02	AC	157 ms	6 MB
g++	boundaryA_03	AC	137 ms	6 MB
g++	boundaryA_04	AC	86 ms	5 MB
g++	boundaryA_05	AC	78 ms	5 MB
g++	boundaryA_06	AC	126 ms	6 MB
g++	boundaryA_07	AC	158 ms	6 MB
g++	boundaryA_08	AC	60 ms	5 MB
g++	boundaryA_09	AC	89 ms	5 MB
g++	boundaryB_00	AC	42 ms	4 MB
g++	boundaryB_01	AC	95 ms	5 MB
g++	boundaryB_02	AC	156 ms	6 MB
g++	boundaryB_03	AC	135 ms	6 MB
g++	boundaryB_04	AC	85 ms	5 MB
g++	boundaryB_05	AC	77 ms	5 MB
g++	boundaryB_06	AC	126 ms	6 MB
g++	boundaryB_07	AC	156 ms	6 MB
g++	boundaryB_08	AC	62 ms	5 MB
g++	boundaryB_09	AC	88 ms	5 MB
g++	example_00	AC	6 ms	4 MB
g++	example_01	AC	6 ms	4 MB
g++	example_02	AC	5 ms	3 MB
g++	handmade_00	AC	5 ms	4 MB
g++	handmade_01	AC	5 ms	4 MB
g++	handmade_02	AC	5 ms	4 MB
g++	handmade_03	AC	5 ms	4 MB
g++	max_00	AC	195 ms	6 MB
g++	max_01	AC	157 ms	6 MB
g++	max_02	AC	167 ms	6 MB
g++	max_03	AC	167 ms	6 MB
g++	max_04	AC	159 ms	6 MB
g++	max_05	AC	156 ms	6 MB
g++	max_06	AC	159 ms	6 MB
g++	max_07	AC	157 ms	6 MB
g++	max_08	AC	158 ms	6 MB
g++	max_09	AC	155 ms	6 MB
g++	random_00	AC	43 ms	4 MB
g++	random_01	AC	96 ms	5 MB
g++	random_02	AC	125 ms	6 MB
g++	random_03	AC	152 ms	6 MB
g++	random_04	AC	86 ms	5 MB
g++	random_05	AC	78 ms	5 MB
g++	random_06	AC	129 ms	6 MB
g++	random_07	AC	155 ms	6 MB
g++	random_08	AC	61 ms	5 MB
g++	random_09	AC	89 ms	5 MB
g++	small_00	AC	6 ms	4 MB
clang++	boundaryA_00	AC	47 ms	4 MB
clang++	boundaryA_01	AC	97 ms	5 MB
clang++	boundaryA_02	AC	146 ms	6 MB
clang++	boundaryA_03	AC	140 ms	6 MB
clang++	boundaryA_04	AC	95 ms	5 MB
clang++	boundaryA_05	AC	85 ms	5 MB
clang++	boundaryA_06	AC	134 ms	6 MB
clang++	boundaryA_07	AC	160 ms	6 MB
clang++	boundaryA_08	AC	66 ms	5 MB
clang++	boundaryA_09	AC	97 ms	5 MB
clang++	boundaryB_00	AC	46 ms	4 MB
clang++	boundaryB_01	AC	104 ms	5 MB
clang++	boundaryB_02	AC	145 ms	6 MB
clang++	boundaryB_03	AC	140 ms	6 MB
clang++	boundaryB_04	AC	139 ms	5 MB
clang++	boundaryB_05	AC	91 ms	5 MB
clang++	boundaryB_06	AC	139 ms	6 MB
clang++	boundaryB_07	AC	160 ms	6 MB
clang++	boundaryB_08	AC	66 ms	5 MB
clang++	boundaryB_09	AC	96 ms	5 MB
clang++	example_00	AC	6 ms	4 MB
clang++	example_01	AC	5 ms	4 MB
clang++	example_02	AC	5 ms	4 MB
clang++	handmade_00	AC	5 ms	4 MB
clang++	handmade_01	AC	5 ms	4 MB
clang++	handmade_02	AC	5 ms	4 MB
clang++	handmade_03	AC	5 ms	4 MB
clang++	max_00	AC	160 ms	6 MB
clang++	max_01	AC	162 ms	6 MB
clang++	max_02	AC	160 ms	6 MB
clang++	max_03	AC	162 ms	6 MB
clang++	max_04	AC	166 ms	6 MB
clang++	max_05	AC	160 ms	6 MB
clang++	max_06	AC	163 ms	6 MB
clang++	max_07	AC	162 ms	6 MB
clang++	max_08	AC	164 ms	6 MB
clang++	max_09	AC	162 ms	6 MB
clang++	random_00	AC	47 ms	4 MB
clang++	random_01	AC	99 ms	5 MB
clang++	random_02	AC	129 ms	6 MB
clang++	random_03	AC	147 ms	6 MB
clang++	random_04	AC	94 ms	5 MB
clang++	random_05	AC	86 ms	5 MB
clang++	random_06	AC	134 ms	6 MB
clang++	random_07	AC	160 ms	6 MB
clang++	random_08	AC	68 ms	5 MB
clang++	random_09	AC	96 ms	5 MB
clang++	small_00	AC	6 ms	4 MB

Back to top page