Extremal order and coincidence of the prime-power functions $f(n)$ and $F(n)$ (Erdős #878)

For $n=\prod_{1\leq i\leq t} p_i^{k_i}$ (factorisation into distinct primes) define $$f(n)=\sum_i p_i^{\ell_i},$$ where for each $i$ the exponent $\ell_i$ is the unique integer with $n\in[p_i^{\ell_i},p_i^{\ell_i+1})$ (so $p_i^{\ell_i}$ is the largest power of $p_i$ not exceeding $n$). Define also $$F(n)=\max \sum_i a_i,$$ the maximum taken over all choices of distinct integers $a_1,\ldots,a_k\leq n$ that are pairwise coprime and such that every prime factor of every $a_i$ is a prime factor of $n$. Several questions are posed. (i) Is it true that for almost all $n$ one has $f(n)=o(n\log\log n)$ and $F(n)\gg n\log\log n$? (ii) Is $$\max_{n\leq x}f(n)\sim \frac{x\log x}{\log\log x}?$$ (iii) Does $\max_{n\leq x}f(n)=\max_{n\leq x}F(n)$ hold for all $x$, or at least all large $x$? (iv) Find an asymptotic formula for the number of $n<x$ with $f(n)=F(n)$. (v) Find an asymptotic formula for $$H(x)=\sum_{n<x}\frac{f(n)}{n},$$ and in particular decide whether $H(x)\ll x\log\log\log\log x$.

Acceptance. This is a proof-shaped problem with several separable targets; a submission must state which it resolves. FULLY RESOLVES (any one part is a complete result): a full written or machine-checkable proof of a named sub-question — the sharp asymptotic $\max_{n\leq x}f(n)\sim x\log x/\log\log x$ for all $x$ (ii); the almost-all lower bound $F(n)\gg n\log\log n$ (the remaining half of (i)); a decision of whether $\max_{n\leq x}f(n)=\max_{n\leq x}F(n)$ holds for all large $x$ (iii); an asymptotic formula for $\#\{n<x:f(n)=F(n)\}$ (iv); or an asymptotic formula for $H(x)$, in particular deciding $H(x)\ll x\log\log\log\log x$ (v). ADVANCES (checkable milestones): narrow Erdős's bracket $x\log\log\log\log x\ll H(x)\ll x\log\log\log x$ on either side with proof; prove $F(n)\sim\tfrac12 n\log\log n$ for almost all $n$, or any nontrivial almost-all bound on $F(n)$; or a certified computation of $f(n),F(n),H(x)$ over a documented range that extends OEIS A339378 and pins down the smallest counterexamples to the coincidence $\max f=\max F$ beyond $x=210$. Deliver the proof/formalisation or the reproducible computation with its certified output.