Problem 878

Erdős Problem #878: declared status 'open'. Formalized: no. 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$$H(x) \ll x\log\log\log\log x?$$ Current best: Erd\H{o}s [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 may be true that, for almost all $n$,$$F(n)\sim \frac{1}{2}n\log\log n.$$Erd\H{o}s 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$$\liminf \frac{1}{x}\sum_{n<x}\frac{f(n)}{n}<\infty.$$Erd\H{o}s [Er84e] proved that$$x\log\log\log\log x\ll H(x) \ll x\log\log\log x.$$See also [879]. Prize: no. OEIS: A339378. Tags: number theory.