An Early Christmas Algebraic Sequence

Number Theory Level 5

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,