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

$$F(n)=\max \sum_{i} a_i$$

where the maximum is taken over all distinct $a_1,\ldots,a_k\leq n$ 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 $n<x$ such that $f(n)=F(n)$. Find an asymptotic formula for

$$H(x)=\sum_{n<x}\frac{f(n)}{n}.$$

Is it true that

$$H(x) \ll x\log\log\log\log x?$$