math/PrimeCount.cpp

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/**
 * Author: Iurii Pustovalov
 * Date: 2022-11-08
 * Description: counting number of primes below N
 * Time: O(N^2/3)
 */
ll prime_pi(const ll N) {
  if (N <= 1) return 0;
  if (N == 2) return 1;
  const int v = sqrt(N);
  int s = (v + 1) / 2;
  vector<int> smalls(s);
  for (int i = 1; i < s; i++) smalls[i] = i;
  vector<int> roughs(s);
  for (int i = 0; i < s; i++) roughs[i] = 2 * i + 1;
  vector<ll> larges(s);
  for (int i = 0; i < s; i++) larges[i] = (N / (2 * i + 1) - 1) / 2;
  vector<bool> skip(v + 1);
  const auto divide = [](ll n, ll d) -> int { return n / d; };
  const auto half = [](int n) -> int { return (n - 1) >> 1; };
  int pc = 0;
  for (int p = 3; p <= v; p += 2)
    if (!skip[p]) {
      int q = p * p;
      if ((ll)q * q > N) break;
      skip[p] = true;
      for (int i = q; i <= v; i += 2 * p) skip[i] = true;
      int ns = 0;
      for (int k = 0; k < s; k++) {
        int i = roughs[k];
        if (skip[i]) continue;
        ll d = (ll)i * p;
        larges[ns] = larges[k] -
                     (d <= v ? larges[smalls[d >> 1] - pc]
                             : smalls[half(divide(N, d))]) +
                     pc;
        roughs[ns++] = i;
      }
      s = ns;
      for (int i = half(v), j = ((v / p) - 1) | 1; j >= p; j -= 2) {
        int c = smalls[j >> 1] - pc;
        for (int e = (j * p) >> 1; i >= e; i--) smalls[i] -= c;
      }
      pc++;
    }
  larges[0] += (ll)(s + 2 * (pc - 1)) * (s - 1) / 2;
  for (int k = 1; k < s; k++) larges[0] -= larges[k];
  for (int l = 1; l < s; l++) {
    ll q = roughs[l];
    ll M = N / q;
    int e = smalls[half(M / q)] - pc;
    if (e < l + 1) break;
    ll t = 0;
    for (int k = l + 1; k <= e; k++)
      t += smalls[half(divide(M, roughs[k]))];
    larges[0] += t - (ll)(e - l) * (pc + l - 1);
  }
  return larges[0] + 1;
}

#line 1 "math/PrimeCount.cpp"
/**
 * Author: Iurii Pustovalov
 * Date: 2022-11-08
 * Description: counting number of primes below N
 * Time: O(N^2/3)
 */
ll prime_pi(const ll N) {
  if (N <= 1) return 0;
  if (N == 2) return 1;
  const int v = sqrt(N);
  int s = (v + 1) / 2;
  vector<int> smalls(s);
  for (int i = 1; i < s; i++) smalls[i] = i;
  vector<int> roughs(s);
  for (int i = 0; i < s; i++) roughs[i] = 2 * i + 1;
  vector<ll> larges(s);
  for (int i = 0; i < s; i++) larges[i] = (N / (2 * i + 1) - 1) / 2;
  vector<bool> skip(v + 1);
  const auto divide = [](ll n, ll d) -> int { return n / d; };
  const auto half = [](int n) -> int { return (n - 1) >> 1; };
  int pc = 0;
  for (int p = 3; p <= v; p += 2)
    if (!skip[p]) {
      int q = p * p;
      if ((ll)q * q > N) break;
      skip[p] = true;
      for (int i = q; i <= v; i += 2 * p) skip[i] = true;
      int ns = 0;
      for (int k = 0; k < s; k++) {
        int i = roughs[k];
        if (skip[i]) continue;
        ll d = (ll)i * p;
        larges[ns] = larges[k] -
                     (d <= v ? larges[smalls[d >> 1] - pc]
                             : smalls[half(divide(N, d))]) +
                     pc;
        roughs[ns++] = i;
      }
      s = ns;
      for (int i = half(v), j = ((v / p) - 1) | 1; j >= p; j -= 2) {
        int c = smalls[j >> 1] - pc;
        for (int e = (j * p) >> 1; i >= e; i--) smalls[i] -= c;
      }
      pc++;
    }
  larges[0] += (ll)(s + 2 * (pc - 1)) * (s - 1) / 2;
  for (int k = 1; k < s; k++) larges[0] -= larges[k];
  for (int l = 1; l < s; l++) {
    ll q = roughs[l];
    ll M = N / q;
    int e = smalls[half(M / q)] - pc;
    if (e < l + 1) break;
    ll t = 0;
    for (int k = l + 1; k <= e; k++)
      t += smalls[half(divide(M, roughs[k]))];
    larges[0] += t - (ll)(e - l) * (pc + l - 1);
  }
  return larges[0] + 1;
}

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