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 <algorithm>
       
 
       
// ---------- copied the Miller-Rabin primality 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;
       
      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;
       
    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 ----------

       
 
       
int main()
       
{
       
  unsigned int limit = 50000000;
       
  std::cin >> limit;
       
 
       
  // count all prime 2n^2-1
       
  unsigned int count = 0;
       
 
       
  // if candidate[n] is false, then n can't be a prime, else there is a possibility
       
  std::vector<bool> candidate(limit + 1, true);
       
 
       
  // generate a few primes, use slower Miller-Rabin algorithm for convenience
       
  const auto MaxSievePrime = std::max<unsigned int>(limit / 50, 10000);
       
  std::vector<unsigned int> smallPrimes;
       
  for (unsigned int p = 3; p <= MaxSievePrime; p++)
       
    if (isPrime(p))
       
      smallPrimes.push_back(p);
       
  // almost all numbers above this threshold have bigger factors (or are primes)
       
  const auto FilterThreshold = 2 * MaxSievePrime;
       
 
       
  // accelerate with OpenMP
       
#define PARALLEL
       
#ifdef  PARALLEL
       
  unsigned int numCores = 0; // 0 => run on all cores, 1,2,3,... => restrict to 1,2,3,... cores
       
#pragma omp parallel for reduction(+:count) num_threads(numCores) schedule(dynamic, 10000)
       
#endif
       
  for (unsigned long long n = 2; n <= limit; n++)
       
  {
       
    // already filtered ?
       
    if (!candidate[n])
       
      continue;
       
 
       
    // calculate 2n^2-1
       
    auto p = 2 * n * n - 1;
       
    // run primality test
       
    if (isPrime(p))
       
    {
       
      count++;
       
      continue;
       
    }
       
 
       
    // p failed the primality test, figure out which (small) prime is a factor of p
       
    // I didn't observe many "new" small prime factors above a certain threshold
       
    if (n < FilterThreshold)
       
      for (auto s : smallPrimes)
       
        if (p > s && p % s == 0) // trial division
       
          for (auto i = n; i <= limit; i += s) // remove all multiples
       
            candidate[i] = false;
       
  }
       
 
       
  // display result
       
  std::cout << count << std::endl;
       
  return 0;
       
}

This solution contains 33 empty lines, 50 comments and 7 preprocessor commands.