Brute force but constrained.
We can check every number from 10 to 9N+1. Why? The number line which grow much faster than the sum of powers of its digits.
For powers of 4s, notice that 94=6561. Multiple this by 5 and you get 32805. This represents 99999 which already exceeds the sum, so any number larger cannot satisfy our condition.
Generate all digit's powers of N to precompute them.
Check numbers from 10 to 10N+2 to see if the sum of powers of the digits equals itself.
Then, sum up the numbers you found.