If $n$ has the given property, denote by $a$ the number of jumps of type $P_k \mapsto P_{k+2}$, and let $b$ be the number of jumps of type $P_k \mapsto P_{k-3}$. Then $a+b = n$ and $n \mid 2a-3b$, which leads to $n \mid 5a$ and $n \mid 5b$. As $0 \le a, b \le n$, we must have either $a = 0$, $b = n$, or $b = 0$, $a = n$, or $5 \mid n$. By only making jumps of type $P_k \mapsto P_{k+2}$, the frog can visit all the vertices if and only if $n$ is odd; making only jumps of type $P_k \mapsto P_{k-3}$, the frog visits all the vertices if and only if $3 \nmid n$. In conclusion, if $n$ has the given property, then either it is not a multiple of $2$, or it is not a multiple of $3$, or it is a multiple of $5$.

Conversely, we have seen that if $6 \nmid n$, then the frog can visit all the vertices by making jumps of a single type.

All that is left is to give an example of a way to choose the frog's jumps in the case when $5 \mid n$.

There are several such examples. One of them is: the frog jumps $P_k \mapsto P_{k+2}$ (mod $5$) if $5 \nmid k$ and $P_k \mapsto P_{k-3}$ (mod $5$) if $5 \mid k$.