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 <vector>
       
#include <set>
       
#include <map>
       
#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;
       
}
       
 
       
// ---------- problem-specific code ----------

       
 
       
// will be 13!

       
unsigned long long factorial = 1;
       
// factorized 13! such such factors[2] = 10; etc.

       
std::map<unsigned int, unsigned int> factors;
       
// all primes where prime-1 is a divisor of 13!

       
std::set<unsigned int> candidates;
       
// all x where phi(x) = 13!

       
std::vector<unsigned long long> results;
       
 
       
// "build" all numbers using the prime factors in "factors"
       
// if number + 1 is prime then store as a candidate

       
void findCandidates(unsigned long long number = 1, unsigned int lastPrime = 1)
       
{
       
  // all primes consumed ?
       
  auto nextPrime = factors.upper_bound(lastPrime);
       
  if (nextPrime == factors.end())
       
  {
       
    // check if successor is prime
       
    if (isPrime(number + 1))
       
      candidates.insert(number + 1);
       
    return;
       
  }
       
 
       
  // process all combinations recursively
       
  auto base     = nextPrime->first;
       
  auto exponent = nextPrime->second;
       
  for (unsigned int i = 0; i <= exponent; i++)
       
  {
       
    findCandidates(number, base);
       
    number *= base;
       
  }
       
}
       
 
       
// look for all number with phi(x) = 13!

       
void search(unsigned long long number = 1, unsigned long long phi = 1,
       
            unsigned int largestPrime = 1)
       
{
       
  // find next prime (or increase the exponent of the latest prime)
       
  auto from = candidates.lower_bound(largestPrime);
       
  for (auto i = from; i != candidates.end(); i++)
       
  {
       
    // current prime
       
    auto current = *i;
       
 
       
    // include one more prime factor
       
    auto nextNumber = number * current;
       
 
       
    // phi(prime)   = prime - 1
       
    auto nextPhi = phi * (current - 1);
       
    // phi(prime^x) = phi(prime^(x-1)) * prime
       
    if (current == largestPrime)
       
      nextPhi += phi; // phi * current = phi * (current - 1) + phi
       
 
       
    // too large ?
       
    if (nextPhi >  factorial)
       
      break;
       
    // match ?
       
    if (nextPhi == factorial)
       
    {
       
      results.push_back(nextNumber);
       
      break;
       
    }
       
 
       
    // keep going if 13! is still reachable
       
    if (factorial % nextPhi == 0)
       
      search(nextNumber, nextPhi, current);
       
  }
       
}
       
 
       
int main()
       
{
       
  unsigned int thirteen = 13;
       
  unsigned int index = 150000;
       
  std::cin >> thirteen >> index;
       
 
       
  // compute 13!
       
  factorial = 1;
       
  for (unsigned int i = 2; i <= thirteen; i++)
       
    factorial *= i;
       
 
       
  // and factorize it
       
  auto reduce = factorial;
       
  for (unsigned int i = 2; i <= thirteen; i++)
       
    while (reduce % i == 0)
       
    {
       
      factors[i]++;
       
      reduce /= i;
       
    }
       
 
       
  // generate all primes whose predecessor is a divisor of 13!
       
  findCandidates();
       
 
       
  // find all numbers with eulerphi() = 13!
       
  search();
       
 
       
  // array is zero-indexed
       
  index--;
       
  // partial sort: only need the 150000th to be at its correct position
       
  auto pos = results.begin() + index;
       
  std::nth_element(results.begin(), pos, results.end());
       
  // (I could have used std::sort as well, but std::nth_element is a tiny bit faster in this case)
       
  std::cout << *pos << std::endl;
       
  return 0;
       
}

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