It is a sum of even numbers, because they almost all all are powers of 2.

The only element of the sum that can be odd is when p = k. This element is prime(0), which is 0 because 0 is not prime.

Thus, the overall sum is necessarily even.

It is easy to check that this sum is strictly larger than 2. Thus, it cannot be a prime number.

What is this conjecture for?

What is the domain of $k$?