:warning: number-theory/sum-of-multiplicative-function.hpp

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#pragma once

#include "math/util.hpp"
#include "math/prime-sieve.hpp"

template <class T>
T SumOfMultiplicativeFunction(long long n, T e, vector<T> prime_sum, function<T(ll, int, ll)> prime_power_value) {
  using ll = long long;
  ll sq = Math::isqrt(n);
  auto ps = PrimeSieve(sq);
  auto dfs = [&](auto dfs, ll x, T f, int p_idx, int r, ll q) -> T {
    ll m = n / x;
    T ret{};
    ll p = ps[p_idx];
    ret += f * prime_power_value(p, r + 1, q * p);
    if (p * p <= m) ret += dfs(dfs, x * p, f, p_idx, r + 1, q * p);
    f *= prime_power_value(p, r, q);
    ret += f * (prime_sum[m <= sq ? m - 1 : prime_sum.size() - x] - prime_sum[p - 1]);
    for (int j = p_idx + 1; j < (int)ps.size(); j++) {
      ll p1 = ps[j];
      if (p1 * p1 > m) break;
      ret += dfs(dfs, x * p1, f, j, 1, p1);
    }
    return ret;
  };

  T ret = prime_sum.back() + e;
  for (int i = 0; i < (int)ps.size(); i++) {
    int p = ps[i];
    ret += dfs(dfs, p, e, i, 1, p);
  }
  return ret;
}
#line 2 "number-theory/sum-of-multiplicative-function.hpp"

#line 2 "math/util.hpp"

namespace Math {
template <class T>
T safe_mod(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  a %= b;
  return a >= 0 ? a : a + b;
}
template <class T>
T floor(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  return a >= 0 ? a / b : (a + 1) / b - 1;
}
template <class T>
T ceil(T a, T b) {
  assert(b != 0);
  if (b < 0) a = -a, b = -b;
  return a > 0 ? (a - 1) / b + 1 : a / b;
}
long long isqrt(long long n) {
  if (n <= 0) return 0;
  long long x = sqrt(n);
  while ((x + 1) * (x + 1) <= n) x++;
  while (x * x > n) x--;
  return x;
}
// return g=gcd(a,b)
// a*x+b*y=g
// - b!=0 -> 0<=x<|b|/g
// - b=0  -> ax=g
template <class T>
T ext_gcd(T a, T b, T& x, T& y) {
  T a0 = a, b0 = b;
  bool sgn_a = a < 0, sgn_b = b < 0;
  if (sgn_a) a = -a;
  if (sgn_b) b = -b;
  if (b == 0) {
    x = sgn_a ? -1 : 1;
    y = 0;
    return a;
  }
  T x00 = 1, x01 = 0, x10 = 0, x11 = 1;
  while (b != 0) {
    T q = a / b, r = a - b * q;
    x00 -= q * x01;
    x10 -= q * x11;
    swap(x00, x01);
    swap(x10, x11);
    a = b, b = r;
  }
  x = x00, y = x10;
  if (sgn_a) x = -x;
  if (sgn_b) y = -y;
  if (b0 != 0) {
    a0 /= a, b0 /= a;
    if (b0 < 0) a0 = -a0, b0 = -b0;
    T q = x >= 0 ? x / b0 : (x + 1) / b0 - 1;
    x -= b0 * q;
    y += a0 * q;
  }
  return a;
}
constexpr long long inv_mod(long long x, long long m) {
  x %= m;
  if (x < 0) x += m;
  long long a = m, b = x;
  long long y0 = 0, y1 = 1;
  while (b > 0) {
    long long q = a / b;
    swap(a -= q * b, b);
    swap(y0 -= q * y1, y1);
  }
  if (y0 < 0) y0 += m / a;
  return y0;
}
long long pow_mod(long long x, long long n, long long m) {
  x = (x % m + m) % m;
  long long y = 1;
  while (n) {
    if (n & 1) y = y * x % m;
    x = x * x % m;
    n >>= 1;
  }
  return y;
}
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 = x % m;
  if (y >= m) y += m;
  while (n) {
    if (n & 1) r = (r * y) % _m;
    y = (y * y) % _m;
    n >>= 1;
  }
  return r;
}
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);
};  // namespace Math
#line 2 "math/prime-sieve.hpp"

vector<int> PrimeSieve(int n) {
  vector<bool> f(n + 1, false);
  for (int p = 2; p * p <= n; p += (p & 1) + 1) {
    if (f[p]) continue;
    for (int i = p * p; i <= n; i += p) f[i] = true;
  }
  vector<int> ps;
  for (int p = 2; p <= n; p += (p & 1) + 1)
    if (!f[p]) ps.push_back(p);
  return ps;
}
/**
 * @brief 素数篩
 */
#line 5 "number-theory/sum-of-multiplicative-function.hpp"

template <class T>
T SumOfMultiplicativeFunction(long long n, T e, vector<T> prime_sum, function<T(ll, int, ll)> prime_power_value) {
  using ll = long long;
  ll sq = Math::isqrt(n);
  auto ps = PrimeSieve(sq);
  auto dfs = [&](auto dfs, ll x, T f, int p_idx, int r, ll q) -> T {
    ll m = n / x;
    T ret{};
    ll p = ps[p_idx];
    ret += f * prime_power_value(p, r + 1, q * p);
    if (p * p <= m) ret += dfs(dfs, x * p, f, p_idx, r + 1, q * p);
    f *= prime_power_value(p, r, q);
    ret += f * (prime_sum[m <= sq ? m - 1 : prime_sum.size() - x] - prime_sum[p - 1]);
    for (int j = p_idx + 1; j < (int)ps.size(); j++) {
      ll p1 = ps[j];
      if (p1 * p1 > m) break;
      ret += dfs(dfs, x * p1, f, j, 1, p1);
    }
    return ret;
  };

  T ret = prime_sum.back() + e;
  for (int i = 0; i < (int)ps.size(); i++) {
    int p = ps[i];
    ret += dfs(dfs, p, e, i, 1, p);
  }
  return ret;
}
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