Question 1
Let the integer sequence $(a_n)_{n\ge0}$ be defined by $a_0=1$ and
$$a_n=a_{n-1}+a_{\lfloor n/3\rfloor}\qquad(n\ge1).$$
Prove that for every prime $p\le13$, there are infinitely many integers $k\ge0$ for which $p\mid a_k$. Here $\lfloor x\rfloor$ is the integer part of $x$.
$$a_n=a_{n-1}+a_{\lfloor n/3\rfloor}\qquad(n\ge1).$$
Prove that for every prime $p\le13$, there are infinitely many integers $k\ge0$ for which $p\mid a_k$. Here $\lfloor x\rfloor$ is the integer part of $x$.