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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 $n<x$ such that $f(n)=F(n)$. Find an asymptotic formula for \[H(x)=\sum_{n<x}\frac{f(n)}{n}.\] Is it true that \[H(x) \ll x\log\log\log\log x?\]

Erdős [Er84e] proved that \[\max_{n\leq x}f(n)\sim \frac{x\log x}{\log\log x}\] for a sequence of $x\to \infty$.

It is trivial that $f(n)\leq F(n)$ for all $n$. It may be true that, for almost all $n$, \[F(n)\sim \frac{1}{2}n\log\log n.\]

Erdős notes that $f(n)/n$ 'almost behaves as a conventional additive function', but unusually $f(n)/n$ does not have a mean value - indeed, \[\limsup \frac{1}{x}\sum_{n<x}\frac{f(n)}{n}=\infty\] but \[\liminf \frac{1}{x}\sum_{n<x}\frac{f(n)}{n}<\infty.\] Erdős [Er84e] proved that \[x\log\log\log\log x\ll H(x) \ll x\log\log\log x.\]