Problem

Source: Romania JBMO TST 2022

Tags: number theory, romania, Romanian TST



Let $p_i$ denote the $i^{\text{th}}$ prime number. For any positive integer $k$ let $a_k$ denote the number of positive integers $t$ such that $p_tp_{t+1}$ divides $k.$ Let $n$ be an arbitrary positive integer. Prove that \[a_1+a_2+\cdots+a_n<\frac{n}{3}.\]