# A series that has prime numbers as a element

I was looking at the following sequence 3,5,7,9,11,13,15,17,19,21,23,...

The terms are given by $a_n=n^2-(n-1)^2$.

When I expanded the sequence I noticed that it contained all the prime numbers between $2$ and $100$

Will the all the prime numbers between 2 and $a_n$ remain a element of the series as $n$ increase?

Yes, but only because every prime number besides $2$ is odd. If we simplify $a_n$ we get $$a_n = n^2-(n-1)^2 = n^2 - (n^2 - 2n + 1) = 2n - 1,$$ meaning $a_n$ is the $n$th odd number.