library

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:heavy_check_mark: test/1_mytest/sigma_0_sum.test.cpp

Depends on

Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include "my_template.hpp"

#include "nt/sigma_0_sum.hpp"

void test() {
  FOR(N, 1, 10000) {
    u64 ans = 0;
    FOR(x, 1, N + 1) ans += N / x;
    assert(ans == sigma_0_sum_small(N));
    assert(ans == sigma_0_sum_large(N));
  }
  u64 N = 1'000'000'000'000'000;
  u64 ANS = 34'693'207'724'724'246; // https://oeis.org/A057494
  u64 a = sigma_0_sum_small(N);
  u64 b = sigma_0_sum_small(N);
  assert(a == ANS && b == ANS);
}

void solve() {
  int a, b;
  cin >> a >> b;
  cout << a + b << "\n";
}

int main() {
  test();
  solve();
}
#line 1 "test/1_mytest/sigma_0_sum.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 1 "my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else

// https://codeforces.com/blog/entry/96344
#pragma GCC optimize("Ofast,unroll-loops")
// いまの CF だとこれ入れると動かない？
// #pragma GCC target("avx2,popcnt")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u8 = uint8_t;
using u16 = uint16_t;
using u32 = uint32_t;
using u64 = uint64_t;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'010'000'000;
template <>
constexpr ll infty<ll> = 2'020'000'000'000'000'000;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * 2'000'000'000'000'000'000;
template <>
constexpr double infty<double> = infty<ll>;
template <>
constexpr long double infty<long double> = infty<ll>;

using pi = pair<ll, ll>;
using vi = vector<ll>;
template <class T>
using vc = vector<T>;
template <class T>
using vvc = vector<vc<T>>;
template <class T>
using vvvc = vector<vvc<T>>;
template <class T>
using vvvvc = vector<vvvc<T>>;
template <class T>
using vvvvvc = vector<vvvvc<T>>;
template <class T>
using pq = priority_queue<T>;
template <class T>
using pqg = priority_queue<T, vector<T>, greater<T>>;

#define vv(type, name, h, ...) vector<vector<type>> name(h, vector<type>(__VA_ARGS__))
#define vvv(type, name, h, w, ...) vector<vector<vector<type>>> name(h, vector<vector<type>>(w, vector<type>(__VA_ARGS__)))
#define vvvv(type, name, a, b, c, ...) \
  vector<vector<vector<vector<type>>>> name(a, vector<vector<vector<type>>>(b, vector<vector<type>>(c, vector<type>(__VA_ARGS__))))

// https://trap.jp/post/1224/
#define FOR1(a) for (ll _ = 0; _ < ll(a); ++_)
#define FOR2(i, a) for (ll i = 0; i < ll(a); ++i)
#define FOR3(i, a, b) for (ll i = a; i < ll(b); ++i)
#define FOR4(i, a, b, c) for (ll i = a; i < ll(b); i += (c))
#define FOR1_R(a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR2_R(i, a) for (ll i = (a)-1; i >= ll(0); --i)
#define FOR3_R(i, a, b) for (ll i = (b)-1; i >= ll(a); --i)
#define overload4(a, b, c, d, e, ...) e
#define overload3(a, b, c, d, ...) d
#define FOR(...) overload4(__VA_ARGS__, FOR4, FOR3, FOR2, FOR1)(__VA_ARGS__)
#define FOR_R(...) overload3(__VA_ARGS__, FOR3_R, FOR2_R, FOR1_R)(__VA_ARGS__)

#define FOR_subset(t, s) for (ll t = (s); t >= 0; t = (t == 0 ? -1 : (t - 1) & (s)))
#define all(x) x.begin(), x.end()
#define len(x) ll(x.size())
#define elif else if

#define eb emplace_back
#define mp make_pair
#define mt make_tuple
#define fi first
#define se second

#define stoi stoll

int popcnt(int x) { return __builtin_popcount(x); }
int popcnt(u32 x) { return __builtin_popcount(x); }
int popcnt(ll x) { return __builtin_popcountll(x); }
int popcnt(u64 x) { return __builtin_popcountll(x); }
int popcnt_mod_2(int x) { return __builtin_parity(x); }
int popcnt_mod_2(u32 x) { return __builtin_parity(x); }
int popcnt_mod_2(ll x) { return __builtin_parityll(x); }
int popcnt_mod_2(u64 x) { return __builtin_parityll(x); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 1, 2)
int topbit(int x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(u32 x) { return (x == 0 ? -1 : 31 - __builtin_clz(x)); }
int topbit(ll x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
int topbit(u64 x) { return (x == 0 ? -1 : 63 - __builtin_clzll(x)); }
// (0, 1, 2, 3, 4) -> (-1, 0, 1, 0, 2)
int lowbit(int x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(u32 x) { return (x == 0 ? -1 : __builtin_ctz(x)); }
int lowbit(ll x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }
int lowbit(u64 x) { return (x == 0 ? -1 : __builtin_ctzll(x)); }

template <typename T>
T floor(T a, T b) {
  return a / b - (a % b && (a ^ b) < 0);
}
template <typename T>
T ceil(T x, T y) {
  return floor(x + y - 1, y);
}
template <typename T>
T bmod(T x, T y) {
  return x - y * floor(x, y);
}
template <typename T>
pair<T, T> divmod(T x, T y) {
  T q = floor(x, y);
  return {q, x - q * y};
}

template <typename T, typename U>
T SUM(const vector<U> &A) {
  T sm = 0;
  for (auto &&a: A) sm += a;
  return sm;
}

#define MIN(v) *min_element(all(v))
#define MAX(v) *max_element(all(v))
#define LB(c, x) distance((c).begin(), lower_bound(all(c), (x)))
#define UB(c, x) distance((c).begin(), upper_bound(all(c), (x)))
#define UNIQUE(x) sort(all(x)), x.erase(unique(all(x)), x.end()), x.shrink_to_fit()

template <typename T>
T POP(deque<T> &que) {
  T a = que.front();
  que.pop_front();
  return a;
}
template <typename T>
T POP(pq<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(pqg<T> &que) {
  T a = que.top();
  que.pop();
  return a;
}
template <typename T>
T POP(vc<T> &que) {
  T a = que.back();
  que.pop_back();
  return a;
}

template <typename F>
ll binary_search(F check, ll ok, ll ng, bool check_ok = true) {
  if (check_ok) assert(check(ok));
  while (abs(ok - ng) > 1) {
    auto x = (ng + ok) / 2;
    (check(x) ? ok : ng) = x;
  }
  return ok;
}
template <typename F>
double binary_search_real(F check, double ok, double ng, int iter = 100) {
  FOR(iter) {
    double x = (ok + ng) / 2;
    (check(x) ? ok : ng) = x;
  }
  return (ok + ng) / 2;
}

template <class T, class S>
inline bool chmax(T &a, const S &b) {
  return (a < b ? a = b, 1 : 0);
}
template <class T, class S>
inline bool chmin(T &a, const S &b) {
  return (a > b ? a = b, 1 : 0);
}

// ? は -1
vc<int> s_to_vi(const string &S, char first_char) {
  vc<int> A(S.size());
  FOR(i, S.size()) { A[i] = (S[i] != '?' ? S[i] - first_char : -1); }
  return A;
}

template <typename T, typename U>
vector<T> cumsum(vector<U> &A, int off = 1) {
  int N = A.size();
  vector<T> B(N + 1);
  FOR(i, N) { B[i + 1] = B[i] + A[i]; }
  if (off == 0) B.erase(B.begin());
  return B;
}

// stable sort
template <typename T>
vector<int> argsort(const vector<T> &A) {
  vector<int> ids(len(A));
  iota(all(ids), 0);
  sort(all(ids), [&](int i, int j) { return (A[i] == A[j] ? i < j : A[i] < A[j]); });
  return ids;
}

// A[I[0]], A[I[1]], ...
template <typename T>
vc<T> rearrange(const vc<T> &A, const vc<int> &I) {
  vc<T> B(len(I));
  FOR(i, len(I)) B[i] = A[I[i]];
  return B;
}

template <typename T, typename... Vectors>
void concat(vc<T> &first, const Vectors &... others) {
  vc<T> &res = first;
  (res.insert(res.end(), others.begin(), others.end()), ...);
}
#endif
#line 3 "test/1_mytest/sigma_0_sum.test.cpp"

#line 1 "nt/convex_floor_sum.hpp"

// f: 凸, 非負, 単調増加を仮定
// above(x,y) : y > f(x)
// slope(x,a,b) : f'(x) >= a/b
// return : sum_[0,N) floor(f(x))
// https://qoj.ac/contest/1195/problem/6188
template <typename U, typename ANS_TYPE, typename F1, typename F2>
ANS_TYPE convex_floor_sum(U N, F1 above, F2 slope) {
  if (N == 0) return 0;
  auto check = [&](U x, U y) -> bool { return x < N && above(x, y); };

  using T = ANS_TYPE;
  auto max_add = [&](auto f, U& a, U& b, U c, U d) -> void {
    auto dfs = [&](auto& dfs, U c, U d) -> void {
      if (!f(a + c, b + d)) return;
      a += c, b += d, dfs(dfs, c + c, d + d);
      if (f(a + c, b + d)) a += c, b += d;
    };
    dfs(dfs, c, d);
  };

  assert(!above(0, 0));
  U x = 0, y = 0;
  max_add([&](U x, U y) -> bool { return !above(x, y); }, x, y, 0, 1);
  ++y;
  T ANS = 2 * (y - 1); //  [0,1) x [1,y)

  auto add_ANS = [&](U& x, U& y, U a, U b) -> void {
    U x0 = x, y0 = y;
    max_add(check, x, y, a, b);
    U n = (x - x0) / a;
    //  (x0,y0) to (x,y)
    ANS += 2 * (y0 - 1) * (x - x0);
    ANS += (x - x0 + 1) * (y - y0 + 1) - (n + 1);
  };

  add_ANS(x, y, 1, 0);
  vc<tuple<U, U, U, U>> SBT;
  SBT.eb(1, 0, 0, 1);
  while (x < N - 1) {
    U a, b, c, d;
    tie(a, b, c, d) = SBT.back();
    if (!check(x + c, y + d)) {
      POP(SBT);
      continue;
    }
    auto f = [&](u64 a, u64 b) -> bool {
      if (x + a >= N) return 0;
      if (above(x + a, y + b)) return 0;
      if (slope(x + a, d, c)) return 0;
      return 1;
    };
    max_add(f, a, b, c, d);
    if (check(x + a + c, y + b + d)) {
      max_add([&](U a, U b) -> bool { return check(x + a, y + b); }, c, d, a, b);
      SBT.eb(a, b, c, d);
      continue;
    }
    add_ANS(x, y, c, d);
  }
  ANS /= T(2);
  return ANS;
}
#line 2 "nt/sigma_0_sum.hpp"

// sum_[1,N] sigma_0(n)
template <typename T = u64>
T sigma_0_sum_small(u64 N) {
  u32 sq = sqrtl(N);
  T ANS = 0;
  for (u32 d = 1; d <= sq; ++d) { ANS += N / d; }
  return 2 * ANS - u64(sq) * sq;
}

// https://oeis.org/A006218
// sigma0(1)+...+sigma0(N) = sum floor(N/i)
template <typename T = u64>
T sigma_0_sum_large(u64 N) {
  u32 sq = sqrtl(N);
  auto above = [&](u128 x, u128 y) -> bool { return y * (sq - x) > N; };
  auto slope = [&](u128 x, u128 a, u128 b) -> bool {
    x = sq - x;
    return a * x * x <= N * b;
  };
  T ANS = convex_floor_sum<u64, T>(sq, above, slope);
  return 2 * ANS - u64(sq) * sq;
}

template <typename T = u64>
T sigma_0_sum(u64 N) {
  return (N < (1e14) ? sigma_0_sum_small<T>(N) : sigma_0_sum_large<T>(N));
}
#line 5 "test/1_mytest/sigma_0_sum.test.cpp"

void test() {
  FOR(N, 1, 10000) {
    u64 ans = 0;
    FOR(x, 1, N + 1) ans += N / x;
    assert(ans == sigma_0_sum_small(N));
    assert(ans == sigma_0_sum_large(N));
  }
  u64 N = 1'000'000'000'000'000;
  u64 ANS = 34'693'207'724'724'246; // https://oeis.org/A057494
  u64 a = sigma_0_sum_small(N);
  u64 b = sigma_0_sum_small(N);
  assert(a == ANS && b == ANS);
}

void solve() {
  int a, b;
  cin >> a >> b;
  cout << a + b << "\n";
}

int main() {
  test();
  solve();
}
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