A natural number $k>1$ is called good if there exist natural numbers $a_1<a_2<\dots <a_k$ such that $\frac{1}{\sqrt{a_1}}+\frac{1}{\sqrt{a_2}}+\dots +\frac{1}{\sqrt{a_k}}=1$. Let $f\left(n\right)$ be the sum of the first $n$ good numbers, $n\ge 1$. Find the sum of all values of $n$ for which $\frac{f\left(n+5\right)}{f\left(n\right)}$ is an integer.