D2.cpp

#include <bits/stdc++.h>
using namespace std;

#define nl '\n'

vector<int> smallest_factor;
vector<bool> prime;
vector<int> primes;
 
void sieve(int maximum) {
    maximum = max(maximum, 1);
    smallest_factor.assign(maximum + 1, 0);
    prime.assign(maximum + 1, true);
    prime[0] = prime[1] = false;
    primes = {};
 
    for (int p = 2; p <= maximum; p++)
        if (prime[p]) {
            smallest_factor[p] = p;
            primes.push_back(p);
 
            for (int64_t i = int64_t(p) * p; i <= maximum; i += p)
                if (prime[i]) {
                    prime[i] = false;
                    smallest_factor[i] = p;
                }
        }
}
 
// Prime factorizes n in worst case O(sqrt n / log n). Requires having run `sieve` up to at least sqrt(n).
// If we've run `sieve` up to at least n, takes O(log n) time.
vector<pair<int64_t, int>> prime_factorize(int64_t n) {
    int64_t sieve_max = int64_t(smallest_factor.size()) - 1;
    assert(1 <= n && n <= sieve_max * sieve_max);
    vector<pair<int64_t, int>> result;
 
    if (n <= sieve_max) {
        while (n != 1) {
            int64_t p = smallest_factor[n];
            int exponent = 0;
 
            do {
                n /= p;
                exponent++;
            } while (n % p == 0);
 
            result.emplace_back(p, exponent);
        }
 
        return result;
    }
 
    for (int64_t p : primes) {
        if (p * p > n)
            break;
 
        if (n % p == 0) {
            result.emplace_back(p, 0);
 
            do {
                n /= p;
                result.back().second++;
            } while (n % p == 0);
        }
    }
 
    if (n > 1)
        result.emplace_back(n, 1);
 
    return result;
}
 
vector<int64_t> generate_factors(const vector<pair<int64_t, int>> &prime_factors, bool sorted = false) {
    // See http://oeis.org/A066150 and http://oeis.org/A036451 for upper bounds on number of factors.
    static vector<int64_t> buffer;
    int product = 1;
 
    for (auto &pf : prime_factors)
        product *= pf.second + 1;
 
    vector<int64_t> factors = {1};
    factors.reserve(product);
 
    if (sorted)
        buffer.resize(product);
 
    for (auto &pf : prime_factors) {
        int64_t p = pf.first;
        int exponent = pf.second;
        int before_size = int(factors.size());
 
        for (int i = 0; i < exponent * before_size; i++)
            factors.push_back(factors[factors.size() - before_size] * p);
 
        if (sorted && factors[before_size - 1] > p)
            for (int section = before_size; section < int(factors.size()); section *= 2)
                for (int i = 0; i + section < int(factors.size()); i += 2 * section) {
                    int length = min(2 * section, int(factors.size()) - i);
                    merge(factors.begin() + i, factors.begin() + i + section,
                          factors.begin() + i + section, factors.begin() + i + length,
                          buffer.begin());
                    copy(buffer.begin(), buffer.begin() + length, factors.begin() + i);
                }
    }
 
    return factors;
}

const int maxN = 4e6 + 10;

void run_cases() {
    int64_t N, P;
    cin >> N >> P;

    vector<int64_t> dp(N + 1);
    dp[1] = 1;

    int64_t sum = 1;

    for(int i = 2; i <= N; i++) {
        sum += dp[i - 1];
        sum %= P;
        sum += dp[1];
        sum %= P;

        for(auto u: generate_factors(prime_factorize(i))) {
            if(u == 1) continue;
            sum += dp[u];
            sum %= P;
            sum -= dp[u - 1];
            sum %= P;
            sum += P;
            sum %= P;
        }
        dp[i] = sum;
        sum += sum;
    }

    cout << dp[N] << '\n';
}

int main() {
    ios_base::sync_with_stdio(0); cin.tie(nullptr);

    sieve(maxN);

    int tests = 1;
    // cin >> tests;

    for(int test = 1;test <= tests;test++) {
        run_cases();
    }
}