Question 103
Let $p_{i}\left(i \in \mathbb{N}^{*}\right)$ be the $i$-th prime number (in increasing order). For every nonzero natural number $k$, denote by $a_{k}$ the number of nonzero natural numbers $i$ with the property that the product $p_{i} p_{i+1}$ divides the number $k$.
If $n$ is a nonzero natural number, show that
$$
a_{1}+a_{2}+\ldots+a_{n}\lt\frac{n}{3}
$$
If $n$ is a nonzero natural number, show that
$$
a_{1}+a_{2}+\ldots+a_{n}\lt\frac{n}{3}
$$