Can you pile the primes?

Can you send your six children home with the same weight of primes in their number bags? - Question: Suppose you want to make two groups with equal sums using the first $N_2$ prime numbers. What is the smallest value of $N_2$ for which you can do this? The answer is three! (Clearly, that wasn’t actually the puzzle.) The first three primes are $2,$ $3,$ and $5,$ and...