# Divisibility Chains

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 $k+a_k+b_k$

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