test/mytest/primesum_mod6.test.cpp

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Last update: 2023-11-09 01:44:55+09:00
Problem: https://judge.yosupo.jp/problem/aplusb

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Code

#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#include "my_template.hpp"
#include "nt/primesum_mod6.hpp"

void test_count() {
  ll LIM = 10000;
  vc<int> A1(LIM), A5(LIM);
  for (auto&& p: primetable(LIM))
    if (p % 6 == 1) { A1[p]++; }
  for (auto&& p: primetable(LIM))
    if (p % 6 == 5) { A5[p]++; }
  A1 = cumsum<int>(A1, 0);
  A5 = cumsum<int>(A5, 0);

  FOR(N, LIM) {
    PrimeSum_Mod_6<int> X(N);
    X.calc_count();
    FOR(K, 1, N + 10) { assert(X[N / K] == mp(A1[N / K], A5[N / K])); }
  }
}

void test_sum() {
  ll LIM = 10000;
  vc<int> A1(LIM), A5(LIM);
  for (auto&& p: primetable(LIM))
    if (p % 6 == 1) { A1[p] += p; }
  for (auto&& p: primetable(LIM))
    if (p % 6 == 5) { A5[p] += p; }
  A1 = cumsum<int>(A1, 0);
  A5 = cumsum<int>(A5, 0);

  FOR(N, LIM) {
    PrimeSum_Mod_6<int> X(N);
    X.calc_sum();
    FOR(K, 1, N + 10) { assert(X[N / K] == mp(A1[N / K], A5[N / K])); }
  }
}

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

signed main() {
  test_count();
  test_sum();
  solve();

  return 0;
}

#line 1 "test/mytest/primesum_mod6.test.cpp"
#define PROBLEM "https://judge.yosupo.jp/problem/aplusb"
#line 1 "my_template.hpp"
#if defined(LOCAL)
#include <my_template_compiled.hpp>
#else
#pragma GCC optimize("Ofast")
#pragma GCC optimize("unroll-loops")

#include <bits/stdc++.h>

using namespace std;

using ll = long long;
using u32 = unsigned int;
using u64 = unsigned long long;
using i128 = __int128;
using u128 = unsigned __int128;
using f128 = __float128;

template <class T>
constexpr T infty = 0;
template <>
constexpr int infty<int> = 1'000'000'000;
template <>
constexpr ll infty<ll> = ll(infty<int>) * infty<int> * 2;
template <>
constexpr u32 infty<u32> = infty<int>;
template <>
constexpr u64 infty<u64> = infty<ll>;
template <>
constexpr i128 infty<i128> = i128(infty<ll>) * infty<ll>;
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;
}
#endif
#line 2 "nt/primetable.hpp"

template <typename T = int>
vc<T> primetable(int LIM) {
  ++LIM;
  const int S = 32768;
  static int done = 2;
  static vc<T> primes = {2}, sieve(S + 1);

  if (done < LIM) {
    done = LIM;

    primes = {2}, sieve.assign(S + 1, 0);
    const int R = LIM / 2;
    primes.reserve(int(LIM / log(LIM) * 1.1));
    vc<pair<int, int>> cp;
    for (int i = 3; i <= S; i += 2) {
      if (!sieve[i]) {
        cp.eb(i, i * i / 2);
        for (int j = i * i; j <= S; j += 2 * i) sieve[j] = 1;
      }
    }
    for (int L = 1; L <= R; L += S) {
      array<bool, S> block{};
      for (auto& [p, idx]: cp)
        for (int i = idx; i < S + L; idx = (i += p)) block[i - L] = 1;
      FOR(i, min(S, R - L)) if (!block[i]) primes.eb((L + i) * 2 + 1);
    }
  }
  int k = LB(primes, LIM + 1);
  return {primes.begin(), primes.begin() + k};
}
#line 2 "nt/primesum.hpp"

/*
N と完全乗法的関数 f の prefix sum 関数 F を与える。
n = floor(N/d) となる n に対する sum_{p <= n} f(p) を計算する。
特に、素数の k 乗和や、mod m ごとでの素数の k 乗和が計算できる。
Complexity: O(N^{3/4}/logN) time, O(N^{1/2}) space.
*/
template <typename T>
struct PrimeSum {
  ll N;
  ll sqN;
  vc<T> sum_lo, sum_hi;
  bool calculated;

  PrimeSum(ll N) : N(N), sqN(sqrtl(N)), calculated(0) {}

  // [1, x] ただし、x = floor(N, i) の形

  T operator[](ll x) {
    assert(calculated);
    return (x <= sqN ? sum_lo[x] : sum_hi[double(N) / x]);
  }

  template <typename F>
  void calc(const F f) {
    auto primes = primetable<int>(sqN);
    sum_lo.resize(sqN + 1);
    sum_hi.resize(sqN + 1);
    FOR3(i, 1, sqN + 1) sum_lo[i] = f(i) - 1;
    FOR3(i, 1, sqN + 1) sum_hi[i] = f(double(N) / i) - 1;
    for (int p: primes) {
      ll pp = ll(p) * p;
      if (pp > N) break;
      int R = min(sqN, N / pp);
      int M = sqN / p;
      T x = sum_lo[p - 1];
      T fp = sum_lo[p] - sum_lo[p - 1];
      for (int i = 1; i <= M; ++i) sum_hi[i] -= fp * (sum_hi[i * p] - x);
      for (int i = M + 1; i <= R; ++i)
        sum_hi[i] -= fp * (sum_lo[N / (double(i) * p)] - x);
      for (int n = sqN; n >= pp; --n) sum_lo[n] -= fp * (sum_lo[n / p] - x);
    }
    calculated = 1;
  }

  void calc_count() {
    calc([](ll x) -> T { return x; });
  }

  void calc_sum() {
    calc([](ll x) -> T {
      ll a = x, b = x + 1;
      if (!(x & 1)) a /= 2;
      if (x & 1) b /= 2;
      return T(a) * T(b);
    });
  }
};
#line 3 "nt/primesum_mod6.hpp"

template <typename T>
struct PrimeSum_Mod_6 {
  ll N;
  ll sqN;

  PrimeSum<T> A, B;
  PrimeSum_Mod_6(ll N) : N(N), sqN(sqrtl(N)), A(N), B(N) {}

  pair<T, T> operator[](ll x) {
    T a = A[x], b = B[x];
    return {(a + b) / T(2), (a - b) / T(2)};
  }

  void calc_count() {
    A.calc([](ll x) -> T { return ((x + 2) / 3 - (x % 6 == 4)); });
    B.calc([](ll x) -> T { return ((x + 5) % 6 <= 3 ? 1 : 0); });
  }

  void calc_sum() {
    A.calc([](ll x) -> T {
      ll n = (x + 2) / 3 - (x % 6 == 4);
      ll k = n / 2;
      if (n % 2 == 0) { return T(6 * k) * T(k); }
      return T(6 * k) * T(k) + T(6 * k + 1);
    });
    B.calc([](ll x) -> T {
      ll n = (x + 2) / 3 - (x % 6 == 4);
      ll k = n / 2;
      if (n % 2 == 0) { return T(-4 * k); }
      return T(-4 * k + 6 * k + 1);
    });
  }
};
#line 4 "test/mytest/primesum_mod6.test.cpp"

void test_count() {
  ll LIM = 10000;
  vc<int> A1(LIM), A5(LIM);
  for (auto&& p: primetable(LIM))
    if (p % 6 == 1) { A1[p]++; }
  for (auto&& p: primetable(LIM))
    if (p % 6 == 5) { A5[p]++; }
  A1 = cumsum<int>(A1, 0);
  A5 = cumsum<int>(A5, 0);

  FOR(N, LIM) {
    PrimeSum_Mod_6<int> X(N);
    X.calc_count();
    FOR(K, 1, N + 10) { assert(X[N / K] == mp(A1[N / K], A5[N / K])); }
  }
}

void test_sum() {
  ll LIM = 10000;
  vc<int> A1(LIM), A5(LIM);
  for (auto&& p: primetable(LIM))
    if (p % 6 == 1) { A1[p] += p; }
  for (auto&& p: primetable(LIM))
    if (p % 6 == 5) { A5[p] += p; }
  A1 = cumsum<int>(A1, 0);
  A5 = cumsum<int>(A5, 0);

  FOR(N, LIM) {
    PrimeSum_Mod_6<int> X(N);
    X.calc_sum();
    FOR(K, 1, N + 10) { assert(X[N / K] == mp(A1[N / K], A5[N / K])); }
  }
}

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

signed main() {
  test_count();
  test_sum();
  solve();

  return 0;
}