You stopped your while loop the wrong way. As it is, if you allow prime_count to be less than or EQUAL to n, then it adds ONE more prime into the sum. And that is one prime too many.
Next, NEVER use sum as a variable. If you do, then your next post here will be to ask in an anguished voice, why the function sum no longer works. Do not use existing function names as variable names.
The following code (based on yours) does now work:
prime_count = prime_count + 1;
Does it agree?