# Divisibility Chains

**Number Theory**Level 5

Define two positive integer sequences \(\{a_n\}\) and \(\{b_n\}\) be defined as \(a_1 < b_1\), \(a_{n+1}=a_n+1\) and \(b_{n+1}=b_n+1\). These two sequences form a **Divisibility Chain** of length \(n\) if \(a_i\mid b_i\) for \(i=1\to n\).

The sequences \(\{a_n\}\) and \(\{b_n\}\) form the longest possible divisibility chain subject to the restriction that \(1 < a_1\le 1000\) and \(1 < b_1 \le 1000\). If this divisibility chain has length \(k\), then find \[k+a_k+b_k\]