The Smarandache prime-digital sequence (SPDS for brevity) is the sequence of primes whose digits are themselves prime.

For example 257 is an element of this sequence because it is prime itself and its digits: 2, 5 and 7 are also prime.

Task
 Show the first 25 SPDS primes.
 Show the hundredth SPDS prime.


See also

 OEIS A019546: Primes whose digits are primes.
 https://www.scribd.com/document/214851583/On-the-Smarandache-prime-digital-subsequence-sequences


#include <iostream>
#include <cstdint>
 
using integer = uint32_t;
 
integer next_prime_digit_number(integer n) {
    if (n == 0)
        return 2;
    switch (n % 10) {
    case 2:
        return n + 1;
    case 3:
    case 5:
        return n + 2;
    default:
        return 2 + next_prime_digit_number(n/10) * 10;
    }
}
 
bool is_prime(integer n) {
    if (n < 2)
        return false;
    if (n % 2 == 0)
        return n == 2;
    if (n % 3 == 0)
        return n == 3;
    if (n % 5 == 0)
        return n == 5;
    constexpr integer wheel[] = { 4,2,4,2,4,6,2,6 };
    integer p = 7;
    for (;;) {
        for (integer w : wheel) {
            if (p * p > n)
                return true;
            if (n % p == 0)
                return false;
            p += w;
        }
    }
}
 
int main() {
    std::cout.imbue(std::locale(""));
    const integer limit = 1000000000;
    integer n = 0, max = 0;
    std::cout << "First 25 SPDS primes:\n";
    for (int i = 0; n < limit; ) {
        n = next_prime_digit_number(n);
        if (!is_prime(n))
            continue;
        if (i < 25) {
            if (i > 0)
                std::cout << ' ';
            std::cout << n;
        }
        else if (i == 25)
            std::cout << '\n';
        ++i;
        if (i == 100)
            std::cout << "Hundredth SPDS prime: " << n << '\n';
        else if (i == 1000)
            std::cout << "Thousandth SPDS prime: " << n << '\n';
        else if (i == 10000)
            std::cout << "Ten thousandth SPDS prime: " << n << '\n';
        max = n;
    }
    std::cout << "Largest SPDS prime less than " << limit << ": " << max << '\n';
    return 0;
}