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>
       
#include <cmath>
       
 
       
// ---------- mulmod, powmod and Miller-Rabin 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;
       
 
       
#ifdef __GNUC__
       
  // based on GCC's 128 bit implementation
       
  return ((unsigned __int128)a * b) % modulo;
       
#endif
       
 
       
  // 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 => simple implementation

       
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 ----------

       
 
       
// much faster ... but still takes a few seconds

       
unsigned long long search(unsigned long long limit, unsigned int multiple)
       
{
       
  std::vector<unsigned long long> found;
       
  // avoid re-allocations for limit = 10^11
       
  if (limit == 100000000000ULL)
       
    found.reserve(26240000);
       
 
       
  // find primes where sigma(prime^2) is divisible by 2017 (same for sigma(prime^3), ...)
       
  for (unsigned long long p = 2; p*p <= limit; p++)
       
  {
       
    if (!isPrime(p))
       
      continue;
       
 
       
    // at least p^2
       
    auto power = p * p;
       
    while (power <= limit)
       
    {
       
      // sigma = (p^q - 1) / (p - 1)
       
      auto sigma = (power*p - 1) / (p - 1);
       
      if (sigma % multiple == 0)
       
        // including all multiples
       
        for (auto i = 1; i*power <= limit; i++)
       
          if (i % p != 0) // avoid the case where i is a multiple of p because then they aren't coprime anymore
       
            found.push_back(i*power);
       
 
       
      // next iteration will exceed limit ?
       
      if (limit / power < p)
       
        break;
       
 
       
      // keep going ...
       
      power *= p;
       
    }
       
  }
       
 
       
  // sum of all matches
       
  unsigned long long result = 0;
       
 
       
  // switch to "optimized" mode if p^2 > limit
       
  size_t sortedSize = 0;
       
 
       
  // sigma(prime) = prime + 1
       
  // => check all numbers 2017k - 1 whether they are prime
       
  for (unsigned long long p = multiple - 1; p <= limit; p += multiple)
       
  {
       
    // only primes ...
       
    if (!isPrime(p))
       
      continue;
       
 
       
    // "optimized" mode:
       
    // collisions can only occur when multiplying p by a smaller prime which must be part of the "found" container
       
    if (sortedSize == 0 && p*p > limit)
       
    {
       
      // sort all solutions so that I can search for elements with std::binary_search
       
      std::sort(found.begin(), found.end());
       
      sortedSize = found.size();
       
    }
       
 
       
    // sigma(p) is a multiple of 2017 and that's also true for all its multiples
       
    for (unsigned long long i = 1; i*p <= limit; i++)
       
    {
       
      // avoid the case where i is a multiple of p because then they aren't coprime anymore
       
      if (i % p == 0)
       
        continue;
       
 
       
      // another solution (which might be a duplicate, though !)
       
      auto current = i * p;
       
 
       
      // if i is small then a collision is impossible
       
      if (i < multiple - 1)
       
      {
       
        result += current;
       
        continue;
       
      }
       
 
       
      // I can detect collisions immediately if p > sqrt(limit) because then p > i
       
      if (sortedSize > 0)
       
      {
       
        if (!std::binary_search(found.begin(), found.begin() + sortedSize, current))
       
          result += current;
       
      }
       
      else
       
        // potential collision
       
        found.push_back(current);
       
    }
       
  }
       
 
       
  // exclude duplicates
       
  std::sort(found.begin(), found.end());
       
  auto duplicates = std::unique(found.begin(), found.end());
       
  // sum of all found numbers (but count duplicates only once)
       
  for (auto i = found.begin(); i != duplicates; i++)
       
    result += *i;
       
 
       
  return result;
       
}
       
 
       
int main()
       
{
       
  unsigned long long limit = 100000000000ULL; // 10^11
       
  unsigned int    multiple = 2017;
       
  std::cin  >> limit >> multiple;
       
  std::cout << search(limit, multiple) << std::endl;
       
  return 0;
       
}

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