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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