Prime digit sums refer to the sum of digits in a number where each digit is a prime number (2, 3, 5, or 7). For example, in the number 237, the sum of the prime digits is 2 + 3 + 7 = 12. This concept can be useful in number theory and digital signal processing. Turn on screen reader support for accessibility, making content more readable for visually impaired users. Prime digit sums can be applied to various mathematical problems where the properties of prime numbers are relevant.

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Prime Digit Sums is a concept where the sum of the digits of a number is checked for primality. For example, if you have a number like 29, the sum of its digits is 11, which is a prime number. Applying this idea to various scenarios can be both intriguing and educational. On a different note, if you're interested in gaming, consider trying " Mod car parking " , which offers enhanced features and customization options for a more immersive experience.

this only does 393 operations instead of 100000 operations at each iteration from 1 to 400000, no timeout

bool isPrime(long n) {
    if (n <= 1)
        return false;
    if (n == 2)
        return true;
    if (n % 2 == 0)
        return false;
    long sqrtN = sqrt(n);
    for (long i = 3; i <= sqrtN; i += 2) {
        if (n % i == 0)
            return false;
    }
    return true;
}

bool SumOfDigits(string number, int k) {
    for (size_t i = 0; i <= number.size() - k; ++i) {
        int sum = 0;
        for (int j = 0; j < k; ++j) {
            sum += (number[i + j] - '0');
        }
        if (!isPrime(sum)) {
            return false;
        }
    }
    return true;
}

string filler(int n, int k) {
    return to_string(static_cast<long>(pow(10, k)) + n).erase(0, 1);
}

void dehaka(vector<long>& result) {
    vector<long> cache(10000);
    vector<pair<pair<int, long>, vector<int>>> D;
    for (int k = 0; k <= 9999; k++) {
        if (SumOfDigits(filler(k, 4), 3) and SumOfDigits(filler(k, 4), 4)) {
            if (k >= 1000) cache[k] = 1;
            vector<int> temp;
            for (int i = 0; i <= 9; i++) {
                string S = filler(i * 1000 + k / 10, 4);
                if (SumOfDigits(S, 3) and SumOfDigits(S, 4) and SumOfDigits(filler(i * 10000 + k, 5), 5)) temp.push_back(i * 1000 + k / 10);
            }
            if (!temp.empty()) D.push_back({ {k, 0}, temp });
        }
    }
    for (int n = 5; n <= 400000; n++) {
        long total = 0;
        for (int i = 0; i < D.size(); i++) {
            long sum = 0;
            for (int K : D[i].second) sum = (sum + cache[K]) % 1000000007;
            D[i].first.second = sum;
            total = (total + sum) % 1000000007;
        }
        if (n == 5) fill(cache.begin(), cache.end(), 0);
        for (int i = 0; i < D.size(); i++) cache[D[i].first.first] = D[i].first.second;
        result.push_back(total);
    }
}

int main()
{
    vector<long> result = { 0, 9, 90, 303, 280 };
    dehaka(result);
    int q, n;
    cin >> q;
    for (int i = 1; i <= q; i++) {
        cin >> n;
        cout << result[n] << '\n';
    }
}

Prime Digit Sums are an intriguing mathematical concept that combines the properties of prime numbers and digital sums. The digital sum of a number is the sum of its individual digits. For example, the digital sum of 123 is 1 + 2 + 3, which equals 6.

Now, let's explore how this concept is relevant to the method mentioned earlier, which involved calculating the initial summation value and then finding the terms that make a difference in the initial value.

In the context of Prime Digit Sums, you can think of the initial summation value as the sum of digits in a given number. For instance, if we have the number 37, the initial summation value would be 3 + 7 = 10. Much like calculating the cost of groceries, where you sum up the prices of individual food items to get the initial cost.

Similarly, when dealing with prime numbers, the initial summation value can be viewed as the sum of the digits in that prime number, just as we calculate the total cost of a meal at a restaurant, including the prices of different food items.

The fascinating part comes when we start looking for terms that make a difference in this initial value, similar to how we might adjust our grocery list based on changing food prices or find ways to reduce our expenses when gas prices are on the rise.

In the method mentioned earlier, the goal was to identify the terms that, when subtracted from the initial value or added to it, resulted in a specific outcome. In the case of Prime Digit Sums, this can be likened to seeking prime numbers whose digital sum meets certain criteria, much like adjusting our expenses based on changing food https://frmenu.org/ and gas prices https://costcogasprice.org/ to meet a budget.

For instance, if we're interested in finding prime numbers where the digital sum is 7, we'd be looking for numbers like 7, 16, 25, and so on. These are primes that, when their digits are summed, equal 7, just as we might adjust our food and gas expenses to meet a total budget.

In summary, Prime Digit Sums offer a captivating intersection of number theory and digital arithmetic, and they share a common thread with the method mentioned earlier in terms of identifying specific terms to alter an initial value, much like managing expenses in the face of fluctuating food and gas prices. Both involve searching for particular numerical properties to achieve a desired outcome, making them engaging topics for those fascinated by the world of numbers and algorithms.