Triple Double Set

Number Theory Level 4

There are triplets of positive integers satisfying $2^{x} + 2^{y} + 2^{z} = 2336$

Evaluate $\displaystyle \sum_{i = 1}^{n} (x_{i} + y_{i} + z_{i} )$

Dedicated to Brian Charlesworth
Brian Charlesworth's challenge - generalize the problem( for odds , evens and primes) , you will be appriciated