Question about the least prime $p$ such that $p+2n$ is also prime.

Question

In the sequence Least prime $p$ such that $p+2n$ is also prime page (A020483) on OEIS, it says:

If $a(n)$ exists, $a(n) < 2n$

What does it mean?

At first it sounds like if both $p$ and $2n+p$ are prime, $p$ must be $< 2n$, which is obviously not true. Then I thought maybe it means if none of the primes $< 2n$ are $a(n)$, then none of the primes $> 2n$ can be $a(n)$ either. But I can’t see how this could be true…

Answer 1

For any integer $n$, $a(n)$ is the smallest $p$ prime such that $p+2n$ is prime.

The statement means: for every $n\in\Bbb N$, if $a(n)$ exists, then it is smaller than $2n$.

It is false for $n=0$ and $n=1$ as $a(0) = 2 > 0$ and $a(1)=3 > 2$ but it seems to work for the other values. Ex:

• $a(2) = 3 < 4$
• $a(3) = 5 < 6$ ...