test/mytest/prime_sum.test.cpp

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Last update: 2024-03-29 11:46:13+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.hpp"
#include "nt/primetable.hpp"
#include "mod/modint.hpp"

void test_count() {
  vc<int> A(1000);
  for (auto&& p: primetable(1000)) { A[p]++; }
  A = cumsum<int>(A, 0);

  FOR(N, 1000) {
    PrimeSum<int> X(N);
    X.calc_count();
    FOR(K, 1, N + 10) { assert(X[N / K] == A[N / K]); }
  }

  vc<ll> TEN(13);
  TEN[0] = 1;
  FOR(i, 12) TEN[i + 1] = TEN[i] * 10;

  ll N = TEN[12];
  PrimeSum<ll> X(N);
  X.calc_count();
  assert(X[TEN[0]] == 0);
  assert(X[TEN[1]] == 4);
  assert(X[TEN[2]] == 25);
  assert(X[TEN[3]] == 168);
  assert(X[TEN[4]] == 1229);
  assert(X[TEN[5]] == 9592);
  assert(X[TEN[6]] == 78498);
  assert(X[TEN[7]] == 664579);
  assert(X[TEN[8]] == 5761455);
  assert(X[TEN[9]] == 50847534);
  assert(X[TEN[10]] == 455052511);
  assert(X[TEN[11]] == 4118054813);
  assert(X[TEN[12]] == 37607912018);
}

void test_sum() {
  vc<int> A(1000);
  for (auto&& p: primetable(1000)) { A[p] += p; }
  A = cumsum<int>(A, 0);

  FOR(N, 1000) {
    PrimeSum<int> X(N);
    X.calc_sum();
    FOR(K, 1, N + 10) { assert(X[N / K] == A[N / K]); }
  }

  vc<ll> TEN(13);
  TEN[0] = 1;
  FOR(i, 12) TEN[i + 1] = TEN[i] * 10;

  using mint = modint998;
  ll N = TEN[12];
  PrimeSum<mint> X(N);
  X.calc_sum();

  auto f = [&](string S) -> mint {
    mint x = 0;
    for (auto&& s: S) { x = x * mint(10) + mint(s - '0'); }
    return x;
  };

  assert(X[TEN[0]] == f("0"));
  assert(X[TEN[1]] == f("17"));
  assert(X[TEN[2]] == f("1060"));
  assert(X[TEN[3]] == f("76127"));
  assert(X[TEN[4]] == f("5736396"));
  assert(X[TEN[5]] == f("454396537"));
  assert(X[TEN[6]] == f("37550402023"));
  assert(X[TEN[7]] == f("3203324994356"));
  assert(X[TEN[8]] == f("279209790387276"));
  assert(X[TEN[9]] == f("24739512092254535"));
  assert(X[TEN[10]] == f("2220822432581729238"));
  assert(X[TEN[11]] == f("201467077743744681014"));
  assert(X[TEN[12]] == f("18435588552550705911377"));
}

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

signed main() {
  cout << fixed << setprecision(15);

  test_count();
  test_sum();
  solve();

  return 0;
}

#line 1 "test/mytest/prime_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 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 3 "test/mytest/prime_sum.test.cpp"

#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 2 "mod/modint_common.hpp"

struct has_mod_impl {
  template <class T>
  static auto check(T &&x) -> decltype(x.get_mod(), std::true_type{});
  template <class T>
  static auto check(...) -> std::false_type;
};

template <class T>
class has_mod : public decltype(has_mod_impl::check<T>(std::declval<T>())) {};

template <typename mint>
mint inv(int n) {
  static const int mod = mint::get_mod();
  static vector<mint> dat = {0, 1};
  assert(0 <= n);
  if (n >= mod) n %= mod;
  while (len(dat) <= n) {
    int k = len(dat);
    int q = (mod + k - 1) / k;
    dat.eb(dat[k * q - mod] * mint::raw(q));
  }
  return dat[n];
}

template <typename mint>
mint fact(int n) {
  static const int mod = mint::get_mod();
  assert(0 <= n && n < mod);
  static vector<mint> dat = {1, 1};
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * mint::raw(len(dat)));
  return dat[n];
}

template <typename mint>
mint fact_inv(int n) {
  static vector<mint> dat = {1, 1};
  if (n < 0) return mint(0);
  while (len(dat) <= n) dat.eb(dat[len(dat) - 1] * inv<mint>(len(dat)));
  return dat[n];
}

template <class mint, class... Ts>
mint fact_invs(Ts... xs) {
  return (mint(1) * ... * fact_inv<mint>(xs));
}

template <typename mint, class Head, class... Tail>
mint multinomial(Head &&head, Tail &&... tail) {
  return fact<mint>(head) * fact_invs<mint>(std::forward<Tail>(tail)...);
}

template <typename mint>
mint C_dense(int n, int k) {
  static vvc<mint> C;
  static int H = 0, W = 0;
  auto calc = [&](int i, int j) -> mint {
    if (i == 0) return (j == 0 ? mint(1) : mint(0));
    return C[i - 1][j] + (j ? C[i - 1][j - 1] : 0);
  };
  if (W <= k) {
    FOR(i, H) {
      C[i].resize(k + 1);
      FOR(j, W, k + 1) { C[i][j] = calc(i, j); }
    }
    W = k + 1;
  }
  if (H <= n) {
    C.resize(n + 1);
    FOR(i, H, n + 1) {
      C[i].resize(W);
      FOR(j, W) { C[i][j] = calc(i, j); }
    }
    H = n + 1;
  }
  return C[n][k];
}

template <typename mint, bool large = false, bool dense = false>
mint C(ll n, ll k) {
  assert(n >= 0);
  if (k < 0 || n < k) return 0;
  if constexpr (dense) return C_dense<mint>(n, k);
  if constexpr (!large) return multinomial<mint>(n, k, n - k);
  k = min(k, n - k);
  mint x(1);
  FOR(i, k) x *= mint(n - i);
  return x * fact_inv<mint>(k);
}

template <typename mint, bool large = false>
mint C_inv(ll n, ll k) {
  assert(n >= 0);
  assert(0 <= k && k <= n);
  if (!large) return fact_inv<mint>(n) * fact<mint>(k) * fact<mint>(n - k);
  return mint(1) / C<mint, 1>(n, k);
}

// [x^d](1-x)^{-n}
template <typename mint, bool large = false, bool dense = false>
mint C_negative(ll n, ll d) {
  assert(n >= 0);
  if (d < 0) return mint(0);
  if (n == 0) { return (d == 0 ? mint(1) : mint(0)); }
  return C<mint, large, dense>(n + d - 1, d);
}
#line 3 "mod/modint.hpp"

template <int mod>
struct modint {
  static constexpr u32 umod = u32(mod);
  static_assert(umod < u32(1) << 31);
  u32 val;

  static modint raw(u32 v) {
    modint x;
    x.val = v;
    return x;
  }
  constexpr modint() : val(0) {}
  constexpr modint(u32 x) : val(x % umod) {}
  constexpr modint(u64 x) : val(x % umod) {}
  constexpr modint(u128 x) : val(x % umod) {}
  constexpr modint(int x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(ll x) : val((x %= mod) < 0 ? x + mod : x){};
  constexpr modint(i128 x) : val((x %= mod) < 0 ? x + mod : x){};
  bool operator<(const modint &other) const { return val < other.val; }
  modint &operator+=(const modint &p) {
    if ((val += p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator-=(const modint &p) {
    if ((val += umod - p.val) >= umod) val -= umod;
    return *this;
  }
  modint &operator*=(const modint &p) {
    val = u64(val) * p.val % umod;
    return *this;
  }
  modint &operator/=(const modint &p) {
    *this *= p.inverse();
    return *this;
  }
  modint operator-() const { return modint::raw(val ? mod - val : u32(0)); }
  modint operator+(const modint &p) const { return modint(*this) += p; }
  modint operator-(const modint &p) const { return modint(*this) -= p; }
  modint operator*(const modint &p) const { return modint(*this) *= p; }
  modint operator/(const modint &p) const { return modint(*this) /= p; }
  bool operator==(const modint &p) const { return val == p.val; }
  bool operator!=(const modint &p) const { return val != p.val; }
  modint inverse() const {
    int a = val, b = mod, u = 1, v = 0, t;
    while (b > 0) {
      t = a / b;
      swap(a -= t * b, b), swap(u -= t * v, v);
    }
    return modint(u);
  }
  modint pow(ll n) const {
    assert(n >= 0);
    modint ret(1), mul(val);
    while (n > 0) {
      if (n & 1) ret *= mul;
      mul *= mul;
      n >>= 1;
    }
    return ret;
  }
  static constexpr int get_mod() { return mod; }
  // (n, r), r は 1 の 2^n 乗根
  static constexpr pair<int, int> ntt_info() {
    if (mod == 120586241) return {20, 74066978};
    if (mod == 167772161) return {25, 17};
    if (mod == 469762049) return {26, 30};
    if (mod == 754974721) return {24, 362};
    if (mod == 880803841) return {23, 211};
    if (mod == 943718401) return {22, 663003469};
    if (mod == 998244353) return {23, 31};
    if (mod == 1045430273) return {20, 363};
    if (mod == 1051721729) return {20, 330};
    if (mod == 1053818881) return {20, 2789};
    return {-1, -1};
  }
  static constexpr bool can_ntt() { return ntt_info().fi != -1; }
};

#ifdef FASTIO
template <int mod>
void rd(modint<mod> &x) {
  fastio::rd(x.val);
  x.val %= mod;
  // assert(0 <= x.val && x.val < mod);
}
template <int mod>
void wt(modint<mod> x) {
  fastio::wt(x.val);
}
#endif

using modint107 = modint<1000000007>;
using modint998 = modint<998244353>;
#line 7 "test/mytest/prime_sum.test.cpp"

void test_count() {
  vc<int> A(1000);
  for (auto&& p: primetable(1000)) { A[p]++; }
  A = cumsum<int>(A, 0);

  FOR(N, 1000) {
    PrimeSum<int> X(N);
    X.calc_count();
    FOR(K, 1, N + 10) { assert(X[N / K] == A[N / K]); }
  }

  vc<ll> TEN(13);
  TEN[0] = 1;
  FOR(i, 12) TEN[i + 1] = TEN[i] * 10;

  ll N = TEN[12];
  PrimeSum<ll> X(N);
  X.calc_count();
  assert(X[TEN[0]] == 0);
  assert(X[TEN[1]] == 4);
  assert(X[TEN[2]] == 25);
  assert(X[TEN[3]] == 168);
  assert(X[TEN[4]] == 1229);
  assert(X[TEN[5]] == 9592);
  assert(X[TEN[6]] == 78498);
  assert(X[TEN[7]] == 664579);
  assert(X[TEN[8]] == 5761455);
  assert(X[TEN[9]] == 50847534);
  assert(X[TEN[10]] == 455052511);
  assert(X[TEN[11]] == 4118054813);
  assert(X[TEN[12]] == 37607912018);
}

void test_sum() {
  vc<int> A(1000);
  for (auto&& p: primetable(1000)) { A[p] += p; }
  A = cumsum<int>(A, 0);

  FOR(N, 1000) {
    PrimeSum<int> X(N);
    X.calc_sum();
    FOR(K, 1, N + 10) { assert(X[N / K] == A[N / K]); }
  }

  vc<ll> TEN(13);
  TEN[0] = 1;
  FOR(i, 12) TEN[i + 1] = TEN[i] * 10;

  using mint = modint998;
  ll N = TEN[12];
  PrimeSum<mint> X(N);
  X.calc_sum();

  auto f = [&](string S) -> mint {
    mint x = 0;
    for (auto&& s: S) { x = x * mint(10) + mint(s - '0'); }
    return x;
  };

  assert(X[TEN[0]] == f("0"));
  assert(X[TEN[1]] == f("17"));
  assert(X[TEN[2]] == f("1060"));
  assert(X[TEN[3]] == f("76127"));
  assert(X[TEN[4]] == f("5736396"));
  assert(X[TEN[5]] == f("454396537"));
  assert(X[TEN[6]] == f("37550402023"));
  assert(X[TEN[7]] == f("3203324994356"));
  assert(X[TEN[8]] == f("279209790387276"));
  assert(X[TEN[9]] == f("24739512092254535"));
  assert(X[TEN[10]] == f("2220822432581729238"));
  assert(X[TEN[11]] == f("201467077743744681014"));
  assert(X[TEN[12]] == f("18435588552550705911377"));
}

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

signed main() {
  cout << fixed << setprecision(15);

  test_count();
  test_sum();
  solve();

  return 0;
}