Prove that every value in the range of the divisor function is the sum of two other numbers in that range.

Is the following statement true or false?Let $\mathbb{N}$ be the set of positive integers. For any $z > 2$, there always exist $x, y < z$ such that:$$f(x) + f(y) = f(z)$$Where the arithmetic function $f(n)$ is defined as:$$f(n) = \prod_{p^k \parallel n} \left( \frac{p^{k+1}-1}{p-1} \right) = \prod_{p^k \parallel n} (\sigma(p^k) )$$with $f(1) = 1$.

It is easily seen that this implies that $f(n)=\sigma (n)$.

Computational check: I have run a computer program to verify this statement for all values of $z$ up to $1,000,000$ (one million), and no counterexample was found. The statement holds true for every single integer in this range.