Motivated by [this question](https://math.stackexchange.com/questions/1810270/prove-that-if-n5n-8n-then-13n-using-induction):

> 
> Is it true that if $n$ is a power of $13$, then $n \mid 5^n + 8^n$ ?
> 
> 
> 

The limited data in [oeis/A045597](https://oeis.org/A045597) seems to suggest it is true.

The converse does not hold. The first counterexample is $n=114413= 13^2 \cdot 677$.

The data also suggests that

> 
> If $n$ is a power of $13$, then $13n$ is the largest power of $13$ that divides $5^n + 8^n$.
> 
> 
> 

Anyone knows proofs or counterexamples for these two observations?