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>
       
 
       
// return (a*b) % modulo

       
unsigned long long mulmod(unsigned long long a, unsigned long long b, unsigned long long modulo)
       
{
       
  // fast path
       
  if (a <= 0xFFFFFFF && b <= 0xFFFFFFF)
       
    return (a * b) % modulo;
       
 
       
  // we might encounter overflows (slow path)
       
  unsigned long long result = 0;
       
  unsigned long long factor = a % modulo;
       
 
       
  // bitwise multiplication
       
  while (b > 0)
       
  {
       
    // b is odd ? a*b = a + a*(b-1)
       
    if (b & 1)
       
    {
       
      result += factor;
       
      if (result >= modulo)
       
        result %= modulo;
       
    }
       
 
       
    // b is even ? a*b = (2*a)*(b/2)
       
    factor <<= 1;
       
    if (factor >= modulo)
       
      factor %= 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)
       
{
       
  // 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 = powmod(x, 2, 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;
       
}
       
 
       
std::vector<unsigned int>       primes;
       
std::vector<unsigned long long> primeSum;
       
 
       
// make sure that at least "num" primes are available in "primes"

       
void morePrimes(unsigned int num)
       
{
       
  if (primes.empty())
       
  {
       
    primes  .reserve(400000);
       
    primeSum.reserve(400000);
       
 
       
    primes.push_back(2);
       
    primes.push_back(3);
       
 
       
    primeSum.push_back(2);
       
  }
       
 
       
  for (auto i = primes.back() + 2; primes.size() <= num; i += 2)
       
  {
       
    bool isPrime = true;
       
    // test against all prime numbers we have so far (in ascending order)
       
    for (auto x : primes)
       
    {
       
      // prime is too large to be a divisor
       
      if (x*x > i)
       
        break;
       
 
       
      // divisible => not prime
       
      if (i % x == 0)
       
      {
       
        isPrime = false;
       
        break;
       
      }
       
    }
       
 
       
    // yes, we have a prime
       
    if (isPrime)
       
      primes.push_back(i);
       
  }
       
 
       
  for (auto i = primeSum.size(); i < primes.size(); i++)
       
    primeSum.push_back(primeSum.back() + primes[i]);
       
}
       
 
       
int main()
       
{
       
  // generate some primes
       
  const unsigned int PrimesPerBatch = 10000;
       
  morePrimes(PrimesPerBatch);
       
 
       
  unsigned int tests;
       
  std::cin >> tests;
       
  while (tests--)
       
  {
       
    unsigned long long last = 1000000;
       
    std::cin >> last;
       
 
       
    unsigned long long best = 2; // highest prime sum
       
    unsigned int maxLength  = 0; // longest chain (must add plus one)
       
 
       
    // all sequences start with surprisingly small prime numbers
       
    // a brute-force search showed that all "good" chains start at 2..131
       
    unsigned int start = 0; // primes[0] = 2
       
    while (primes[start] <= 131 && primes[start] <= last)
       
    {
       
      unsigned long long subtract = 0;
       
      if (start > 0)
       
        subtract = primeSum[start - 1];
       
 
       
      unsigned int pos = start + maxLength;
       
      // find shortest chain whose sum exceeds the limit
       
      while (primeSum[pos] - subtract <= last)
       
      {
       
        pos++;
       
        // running out of prime numbers ? add more !
       
        if (pos + 100 >= primes.size()) // plus 100 is probably too cautious
       
          morePrimes(primes.size() + PrimesPerBatch);
       
      }
       
      pos--;
       
 
       
      // chop off one prime number until the sum is prime, too
       
      while (pos - start > maxLength)
       
      {
       
        unsigned long long sum = primeSum[pos] - subtract;
       
        // yes, we have a good candidate (maybe better ones for other values of "start", though)
       
        if (isPrime(sum))
       
        {
       
          maxLength = pos - start;
       
          best = sum;
       
          break;
       
        }
       
 
       
        pos--;
       
      }
       
 
       
      start++;
       
    }
       
 
       
    // if sum is > 0 then "length" didn't count the first element
       
    if (best >= 2)
       
      maxLength++;
       
    std::cout << best << " " << maxLength << std::endl;
       
  }
       
 
       
  return 0;
       
}

This solution contains 38 empty lines, 46 comments and 2 preprocessor commands.