Going From Large To Small

Consider infinite sums of reciprocals

$\displaystyle \sum _{ n=1 }^{ \infty }{ \dfrac { 1 }{ { a }_{ n } } }$

where ${a}_{n}$ are the terms in any of the following infinite sets

1) $\;\; (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,\ldots)=n$
2) $\;\; (1, 5, 9, 13, 17, 21, 25, 29, 33, 37, 41, 45,\ldots) = 4n+1$
3) $\;\; (2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,\ldots) = n$ th prime number
4) $\;\; (2, 3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71,\ldots) =$ after $2, 3$ , primes of form $4m+3$
5) $\;\; (3, 5, 5, 7, 11, 13, 17, 19, 29, 31, 41, 43,\ldots)=$ consecutive twin primes
6) $\;\; (1, 10, 11, 100, 101, 102, 1000, 1001, 1002, 1003, 10000, 10001,\ldots)$
7) $\;\; (5, 7, 11, 13, 11, 13, 17, 19, 101, 103, 107, 109,\ldots)=$ consecutive prime quadruplets

Twin Primes are pairs of primes of the form $\; (p, p+2)$
Prime Quadruplets are sets of four primes of the form $\; (p, p+2, p+6, p+8)$

All of these may or may not include some terms (not necessarily consecutive) that are in an arithmetic progression of any arbitrary* or infinite length ( both of which are not the same thing! ), but none of the full sequences are an arithmetic-geometric progression with a common ratio $r>1$ .

For your convenience, the numerical values of each for the first 12 terms are

1) $\;\; 3.10321\ldots$
2) $\;\; 1.67290\ldots$
3) $\;\; 1.59272\ldots$
4) $\;\; 1.28596\ldots$
5) $\;\; 1.26989\ldots$
6) $\;\; 1.22481\ldots$
7) $\;\; 0.82811\ldots$

Some of them may or may not be divergent (although most of you will recognize 1) as being famously divergent), and some of them may or may not be convergent. In any case, the two kinds have been sorted, with all the divergent ones on top and all the convergent ones, if any, on bottom.

Which is the first series listed here that is convergent , going from top 1) to bottom 7) ?

Enter the list number as your answer. If "none", enter 0.

Note: Any arbitrary [integer] means any integer which is not infinite. So, for example, adding any arbitrary integer number of finite terms will always result in a finite total. Whereas, $\infty$ is not an integer number.