My code

… was written in C++11 and can be compiled with G++, Clang++, Visual C++. You can download it, too. Or just jump to my GitHub repository.

       
#include <iostream>
       
#include <string>
       
#include <vector>
       
#include <set>
       
#include <algorithm>
       
 
       
// ---------- Miller-Rabin prime test from my toolbox ----------

       
 
       
// return (a*b) % modulo

       
unsigned long long mulmod(unsigned long long a, unsigned long long b, unsigned long long modulo)
       
{
       
  // (a * b) % modulo = (a % modulo) * (b % modulo) % modulo
       
  a %= modulo;
       
  b %= modulo;
       
 
       
  // fast path
       
  if (a <= 0xFFFFFFF && b <= 0xFFFFFFF)
       
    return (a * b) % modulo;
       
 
       
  // we might encounter overflows (slow path)
       
  // the number of loops depends on b, therefore try to minimize b
       
  if (b > a)
       
    std::swap(a, b);
       
 
       
  // bitwise multiplication
       
  unsigned long long result = 0;
       
  while (a > 0 && b > 0)
       
  {
       
    // b is odd ? a*b = a + a*(b-1)
       
    if (b & 1)
       
    {
       
      result += a;
       
      if (result >= modulo)
       
        result -= modulo;
       
      // skip b-- because the bit-shift at the end will remove the lowest bit anyway
       
    }
       
 
       
    // b is even ? a*b = (2*a)*(b/2)
       
    a <<= 1;
       
    if (a >= modulo)
       
      a -= modulo;
       
 
       
    // next bit
       
    b >>= 1;
       
  }
       
 
       
  return result;
       
}
       
 
       
// return (base^exponent) % modulo

       
unsigned long long powmod(unsigned long long base, unsigned long long exponent, unsigned long long modulo)
       
{
       
  unsigned long long result = 1;
       
  while (exponent > 0)
       
  {
       
    // fast exponentation:
       
    // odd exponent ? a^b = a*a^(b-1)
       
    if (exponent & 1)
       
      result = mulmod(result, base, modulo);
       
 
       
    // even exponent ? a^b = (a*a)^(b/2)
       
    base = mulmod(base, base, modulo);
       
    exponent >>= 1;
       
  }
       
  return result;
       
}
       
 
       
// Miller-Rabin-test

       
bool isPrime(unsigned long long p)
       
{
       
  // IMPORTANT: requires mulmod(a, b, modulo) and powmod(base, exponent, modulo)
       
 
       
  // some code from             https://ronzii.wordpress.com/2012/03/04/miller-rabin-primality-test/
       
  // with optimizations from    http://ceur-ws.org/Vol-1326/020-Forisek.pdf
       
  // good bases can be found at http://miller-rabin.appspot.com/
       
 
       
  // trivial cases
       
  const unsigned int bitmaskPrimes2to31 = (1 <<  2) | (1 <<  3) | (1 <<  5) | (1 <<  7) |
       
                                          (1 << 11) | (1 << 13) | (1 << 17) | (1 << 19) |
       
                                          (1 << 23) | (1 << 29); // = 0x208A28Ac
       
  if (p < 31)
       
    return (bitmaskPrimes2to31 & (1 << p)) != 0;
       
 
       
  if (p %  2 == 0 || p %  3 == 0 || p %  5 == 0 || p % 7 == 0 || // divisible by a small prime
       
      p % 11 == 0 || p % 13 == 0 || p % 17 == 0)
       
    return false;
       
 
       
  if (p < 17*19) // we filtered all composite numbers < 17*19, all others below 17*19 must be prime
       
    return true;
       
 
       
  // test p against those numbers ("witnesses")
       
  // good bases can be found at http://miller-rabin.appspot.com/
       
  const unsigned int STOP = 0;
       
  const unsigned int TestAgainst1[] = { 377687, STOP };
       
  const unsigned int TestAgainst2[] = { 31, 73, STOP };
       
  const unsigned int TestAgainst3[] = { 2, 7, 61, STOP };
       
  // first three sequences are good up to 2^32
       
  const unsigned int TestAgainst4[] = { 2, 13, 23, 1662803, STOP };
       
  const unsigned int TestAgainst7[] = { 2, 325, 9375, 28178, 450775, 9780504, 1795265022, STOP };
       
 
       
  // good up to 2^64
       
  const unsigned int* testAgainst = TestAgainst7;
       
  // use less tests if feasible
       
  if (p < 5329)
       
    testAgainst = TestAgainst1;
       
  else if (p < 9080191)
       
    testAgainst = TestAgainst2;
       
  else if (p < 4759123141ULL)
       
    testAgainst = TestAgainst3;
       
  else if (p < 1122004669633ULL)
       
    testAgainst = TestAgainst4;
       
 
       
  // find p - 1 = d * 2^j
       
  auto d = p - 1;
       
  d >>= 1;
       
  unsigned int shift = 0;
       
  while ((d & 1) == 0)
       
  {
       
    shift++;
       
    d >>= 1;
       
  }
       
 
       
  // test p against all bases
       
  do
       
  {
       
    auto x = powmod(*testAgainst++, d, p);
       
    // is test^d % p == 1 or -1 ?
       
    if (x == 1 || x == p - 1)
       
      continue;
       
 
       
    // now either prime or a strong pseudo-prime
       
    // check test^(d*2^r) for 0 <= r < shift
       
    bool maybePrime = false;
       
    for (unsigned int r = 0; r < shift; r++)
       
    {
       
      // x = x^2 % p
       
      // (initial x was test^d)
       
      x = mulmod(x, x, p);
       
      // x % p == 1 => not prime
       
      if (x == 1)
       
        return false;
       
 
       
      // x % p == -1 => prime or an even stronger pseudo-prime
       
      if (x == p - 1)
       
      {
       
        // next iteration
       
        maybePrime = true;
       
        break;
       
      }
       
    }
       
 
       
    // not prime
       
    if (!maybePrime)
       
      return false;
       
  } while (*testAgainst != STOP);
       
 
       
  // prime
       
  return true;
       
}
       
 
       
// ---------- and now my solution ----------

       
 
       
// a sqube has value = p^2 * q^3

       
struct Sqube
       
{
       
  // note: this struct doesn't check whether p and q are different primes
       
  const unsigned int p;
       
  const unsigned int q;
       
  const unsigned long long value;
       
 
       
  // create a new sqube
       
  Sqube(unsigned int p_, unsigned int q_)
       
  : p(p_), q(q_), value((unsigned long long)p_*p_ * q_*q_*q_)
       
  {}
       
 
       
  // sort two squbes by their value, needed by std::set
       
  bool operator<(const Sqube& other) const
       
  {
       
    return value < other.value;
       
  }
       
};
       
 
       
// return true if changing a digit converts the number to a prime number

       
bool isPrimeProof(unsigned long long value)
       
{
       
  auto strValue = std::to_string(value);
       
  for (unsigned int pos = 0; pos < strValue.size(); pos++)
       
  {
       
    // an even number can only become prime when modifying the last digit
       
    if (value % 2 == 0)
       
      pos = strValue.size() - 1;
       
 
       
    // change digit by digit
       
    auto strModified = strValue;
       
    for (auto digit = '0'; digit <= '9'; digit++)
       
    {
       
      // no leading zero
       
      if (digit == '0' && pos == 0)
       
        continue;
       
      // last digit can't be even
       
      if (digit % 2 == 0 && pos == strValue.size() - 1) // ASCII codes of even digits are even, too
       
        digit++;                                        // strictly speaking this doesn't test 2 (which is a prime)
       
                                                        // but the next number 3 is prime and produced the correct result
       
 
       
      // no need to check the original value (a sqube is never prime)
       
      if (digit == strValue[pos])
       
        continue;
       
 
       
      // convert from string to binary
       
      strModified[pos] = digit;
       
      auto modified = std::stoull(strModified);
       
 
       
      // is it prime ?
       
      if (isPrime(modified))
       
        return false;
       
    }
       
  }
       
 
       
  return true;
       
}
       
 
       
int main()
       
{
       
  // count how many squbes contain "200"
       
  unsigned int sequence = 200;
       
  std::cin >> sequence;
       
 
       
  std::string  strSequence = std::to_string(sequence); // = "200"
       
  unsigned int count       = 0; // stop when count = 200
       
 
       
  // the two smallest squbes, my "seed values"
       
  std::set<Sqube> squbes = { Sqube(3, 2), Sqube(2, 3) };
       
 
       
  while (true) // abort/exit condition can be found inside the loop
       
  {
       
    // pick smallest sqube and remove it
       
    auto current = *(squbes.begin());
       
    squbes.erase(squbes.begin());
       
 
       
    // does it contain "200" ?
       
    auto strCurrent = std::to_string(current.value);
       
    if (strCurrent.find(strSequence) != std::string::npos &&
       
        isPrimeProof(current.value))
       
    {
       
      // yes, another match
       
      count++;
       
 
       
      // done ?
       
      if (count == sequence)
       
      {
       
        std::cout << strCurrent << std::endl;
       
        break;
       
      }
       
    }
       
 
       
    // add next squbes
       
    // find a sqube with the same q but p is the next prime (not equal to q)
       
    auto nextP = current.p + 1;
       
    while (nextP == current.q || !isPrime(nextP))
       
      nextP++;
       
    squbes.insert(Sqube(nextP, current.q));
       
 
       
    // find a sqube with the same p but q is the next prime (not equal to p)
       
    auto nextQ = current.q + 1;
       
    while (nextQ == current.p || !isPrime(nextQ))
       
      nextQ++;
       
    squbes.insert(Sqube(current.p, nextQ));
       
  }
       
 
       
  return 0;
       
}

This solution contains 42 empty lines, 61 comments and 5 preprocessor commands.