Permutation of first $k$ prime powers as a one-way function?
As this stands this is a hard to evaluate and inefficient proposal. 

Firstly key size is effectively  $\log n\!=O(n (\log n-1))$ but the largest value (attained for the identity permutation) is $\sum_{i\leq k} p_i^{p_i}$
which is larger than 

$$(n \log n)^{n \log n}=O(e^{n (\log n)^2}).$$

You would need to look at reduction modulo so elarge integer to make it efficient.

While it may be that the type of diophantine equation that needs to be solvable (for prime arguments) for the original function to have collisions is unsolvable, such things are almost impossible to prove. I am not an expert but I know of no such results even for few terms. 

And once you do a modular reduction, all bets are off.

Computationally, you probably cannot check much beyond the first 20 or so primes for no collisions.

One idea is to check the differences of the values the set of functions for a fixed $k$ give, since there are quadratically many differences,  they’d be more likely to repeat. If this happens this means that you might be able to find a modular reduction that will yield a collision.