Sequence with prime number

Suppose $p$ is a prime number greater than 2, and $m$ is a natural number. Let $a_n$ be sequences defined by:
$a_1 = 1$
$a_2 = m$
$a_{n+2} = \frac{a_{n+1}^2 +p}{a_n}, n = 1,2,...$

Determine all values of $m$ such that $a_n$ is an integer for all $n$.