vector<int> sieve_linear(int n) {
    vector<bool> is_prime(n + 1, true);
    vector<int> primes;

    for (int i = 2; i <= n/2; i++) {
        if (is_prime[i])
            primes.push_back(i);

        for (int j = 0; j < primes.size() && i * primes[j] <= n; j++) {
            is_prime[i * primes[j]] = 0;
            if (i % primes[j] == 0)
                break;
        }
    }

    for (int i = n/2 + 1; i <= n; i++) {
        if (is_prime[i])
            primes.push_back(i);
    }

    return primes;
}