# An Early Christmas Algebraic Sequence

Consider the sequence $${ a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 },{ a }_{ 4 },\dots$$ such that $${ a }_{ 1 }=2$$ and for every positive integer $$n$$,

$${ a }_{ n+1 }={ a }_{ n }+{ p }_{ n }$$ , where $${ p }_{ n }$$ is the largest prime factor of $${ a }_{ n }$$.

The first few terms of the sequence are $$2,4,6,9,12,15,20$$. What is the largest value of $$n$$ such that $${ a }_{ n }$$ is a four-digit number?

Note: This is adapted from a Mathematics competition. Happy solving; From Issac Newton,

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