If $n={\prod }_{1\le i\le t}{p}_i^{k_i}$ 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 $a_1,\dots ,a_k\le n$ such that $(a_i,a_j)=1$ for $i\ne j$ 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$$\underset{n\le x}{\max }f(n)\sim \frac{x\log x}{\log \log x}?$$Is it true that (for all $x$, or perhaps just for all large $x$)$$\underset{n\le x}{\max }f(n)=\underset{n\le x}{\max }F(n)?$$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$$H(x)\ll x\log \log \log \log x?$$