VN-TST 2001 Problem 1

Let a sequence of integers $\{a_n\}$ , $n \in \mathbb{N}$ be given, defined by

$$a_0 = 1, a_n= a_{n-1} + a_{[n/3]}$$

for all $n \in \mathbb{N}^{*}$ .

Show that for all primes $p \leq 13$ , there are infinitely many integer numbers $k$ such that $a_k$ is divided by $p$ . (Here $[x]$ denotes the integral part of real number $x$ ).