An Early Christmas Algebraic Sequence

Consider the sequence a1,a2,a3,a4,{ a }_{ 1 },{ a }_{ 2 },{ a }_{ 3 },{ a }_{ 4 },\dots such that a1=2{ a }_{ 1 }=2 and for every positive integer nn,

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

The first few terms of the sequence are 2,4,6,9,12,15,202,4,6,9,12,15,20. What is the largest value of nn such that an{ a }_{ n } is a four-digit number?

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

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