Let $a_1$ through $a_k$ be some permutation of the first $k$ primes. Let $n \in [1,k!]$ be a parameter specifying the exact ordering by taking the $n$th permutation in a sorted list or by some other simple mechanism.
Then define $f(n,k)=a_1^{p_1}+a_2^{p_2}+\ldots+a_k^{p_k}$, where $p_i$ is the $i$th prime, and the $a_i$ are the permutation of the primes indicated by $n$.
e.g. $f(5,4)=2^2+7^3+3^5+5^7=78715$.
I conjectured that all such sums may be distinct. Supposing that were true, this appears like it could potentially serve as a bijective one-way function using a suitably high value of $k$ and an effectively randomized permutation.
For very small $k$ (up to $20$ or so), the $f(n,k)$ sums are spread far apart and have little opportunity for individual terms to overlap, making it easy to greedily home in on the correct inputs. Beyond that, it quickly becomes more difficult. Barring some tidy solution that I cannot see, finding $f^{-1}(f(n,k))$ seems totally intractable once it's extended to $k>1000$ or more, while $f(n,k)$ is trivial to calculate to very large values. Accordingly, my question:
Could $f(n,k)$ conceivably make a suitable one-way function?
In particular, I'm looking for advice on whether there is likely to be some way to efficiently find $f^{-1}(f(n,k))$ after all, and also whether $f$ looks like it has other properties that would make it either a strong or poor choice for this purpose.