$$\sum_{k=1}^{n} \frac{1}{k}=H_n=\frac{p_n}{q_n}$$
Where $p_n,q_n$ are coprime intergers.
The first few values for $p_n+q_n$ are $2,5,7,37,197,69,504,1041,9649$.
When are $p_n+q_n$ Fibonacci numbers?
Due to the inequality, $\ln(n+1)<H_n<\ln(n)+1$, $\ln(n+1)q_n<p_n<(\ln(n)+1)q_n$.
Therefore, this would imply that $(\ln(n+1)+1)q_n<p_n+q_n<(\ln(n)+2)q_n$.
However, $\frac{\ln(n)+2}{\ln(n+1)+1}<\phi$ where $\phi$ is the golden ratio.
Though this is sloppy logic, because of this I thought it was likely are few cases where $p_n+q_n$ are Fibonacci numbers. The only solutions appeared to be $2,5$.
Any help would be appreciated.