The optimization problem is as follows: \begin{equation} \begin{aligned} &\max\limits_{\boldsymbol{\omega}} \frac{\sum\limits_{n=1}^N \frac{\omega_n k_n}{k_n-1}}{\sqrt{\sum\limits_{n=1}^N \omega_n^2 \frac{k_n^3}{(k_n-1)^2(k_n-2)}}}\\ &\text{s.t.}\sum\limits_{n=1}^N\omega_n=1 \end{aligned} \end{equation} where $\boldsymbol{\omega}=\left[\omega_1,\omega_2,\cdots,\omega_N \right], 0\leq\omega_n\leq1$, and $k_n$ are integers such that $k_1 \neq k_2\cdots \neq k_N$.
Problem: Can we get closed-from solutions for this optimization problem?
Too long for a comment.
Calling $a_i = \frac{k_i}{k_i-1}$ and $b_i = \frac{k_i^3}{(k_i-1)^2(k_i-2)}$ the problem is equivalent to
$$ \max_{\omega}\frac{(a\cdot\omega)^2}{\sum b_i\omega_i^2}\ \ \text{s. t.}\ \ \cases{\sum \omega_i = 1\\ \ 0\le\omega_i\le 1} $$
The lagrangian
$$ L(\omega,\lambda) = \frac{(a\cdot\omega)^2}{\sum b_i\omega_i^2}+\lambda\left(\sum \omega_i - 1\right) $$
has stationary points at
$$ \cases{ 2\frac{(a\cdot\omega)a_i}{\sum b_i\omega_i^2}-2\frac{(a\cdot\omega)^2\omega_i b_i}{\left(\sum b_i\omega_i^2\right)^2}+\lambda=0\\ \sum \omega_i = 1} $$
or
$$ \cases{ (\sum b_i\omega_i^2)a_i-(a\cdot\omega)\omega_i b_i+\frac 12\frac{(\sum b_i\omega_i^2)^2}{a\cdot\omega}\lambda=0\\ \sum \omega_i = 1} $$
or as $\lambda$ is generic, we follow with $\mu = \frac 12\frac{(\sum b_i\omega_i^2)^2}{a\cdot\omega}\lambda$
$$ \cases{ (\sum b_i\omega_i^2)a_i-(a\cdot\omega)\omega_i b_i+\mu=0\\ \sum \omega_i = 1} $$
now multiplying the line $i$ by $\omega_i$ and adding all lines we can conclude that $\mu = 0$
NOTE
This implies on
$$ \frac{\omega_i b_i}{a_i}=\kappa,\ \forall i $$
and finally
$$ \omega_i = \frac{\frac{a_i}{b_i}}{\sum\frac{a_i}{b_i}} $$
so we have also $0\le \omega_i\le 1,\ \ \forall i$