Let $(p_{n})$ be the sequences of primes. We know that the quantity $g_{n}=p_{n+1}-p_{n}$ is called gap. Consider the case of twin primes $p_{n+1}-p_{n}=2$.
My question is:
What is the name of rank $n$ where these primes occurs. For example, $n=2,3,...$.
The general term, for the numerical subscript when used in mathematics related to sets, is an index. Its used as reference to the ordering of the elements of the ordered set (or a sequence more likely). For example if my set were $\{2x:x\in\mathbb{N}\}$ (aka the even numbers) then then $x$ acts as an index and arguably could be referred to for the $x$th even number by using notation as $2_x$, however dangerous (definition might be required at that point though).
Another way subscripts are used in mathematics,and computing, is for bases. A numerical string, could be $123_3$ for $1\cdot 3^2 +2\cdot 3+3$ for example.
You can index in two dimensions as well, like binomial coefficients in pascal's triangle, or rows and columns of a matrix. It just takes two reference points though, usually using a hanging subscript or a comma in a subscript $_nC_r$ (sometimes written $^nC_r$) and $H_{n,m}$ for example.
EDIT we could call them the set of numbers that aren't of form $2ab+a+b$, because $2(2ab+a+b)+1 = (2a+1)(2b+1)$ which makes it composite if $a,b>0$ via Sieve of Sundaram., okay thinking wrong way, it's just an index on the primes which are a set of numbers set out as above.