Erdős Problem 878

### What is the conjecture

https://www.erdosproblems.com/878

If $n=\prod_{1\leq i\leq t} p_i^{k_i}$ is the factorisation of $n$ into distinct primes then let
$$f(n)=\sum p_i^{\ell_i},$$
where $\ell_i$ is chosen such that $n\in [p_i^{\ell_i},p_i^{\ell_i+1})$. Furthermore, let
$$F(n)=\max \sum_{i=1}^t a_i$$
where the maximum is taken over all $a_1,\ldots,a_t\leq n$ such that $(a_i,a_j)=1$ for $i\neq j$ and all prime factors of each $a_i$ are prime factors of $n$.

Is it true that, for almost all $n$,
$$f(n)=o(n\log\log n)$$
and
$$F(n) \gg n\log\log n?$$
Is it true that
$$\max_{n\leq x}f(n)\sim \frac{x\log x}{\log\log x}?$$
Is it true that (for all $x$, or perhaps just for all large $x$)
$$\max_{n\leq x}f(n)=\max_{n\leq x}F(n)?$$
Find an asymptotic formula for the number of $n