A Formula For Primes Could someone explain me why this arithmetic of sets can not be called a Prime Numbers formula? Was it already found before and is not relevant?
Prime numbers sequence $\mathbb P$ (or set of members expressed as a sequence) is revealed from a combination of $5$ sets (where $k,i,j\in \mathbb N$)


*

*Let the set $\{a_k\}$ be defined by $a_k=6k-1$. 

*Let the set $\{b_k\}$ be defined by $b_k=6k+1$.

*Define the sets $\{o_{1,i,j}\}$ and $\{o_{2,i,j}\}$ and $\{o_{3,i,j}\}$ be defined as $o_{1,i,j}=(6i-1)(6j+1)$ and $o_{2,i,j}=(6i-1)(6j-1)$ and $o_{3,i,j}=(6i+1)(6j+1)$.


Then, we can write the set of primes as
$$\mathbb P = \{a_k\}  \cup \{b_k\} \setminus \{o_{1,i,j}\} \setminus \{o_{2,i,j}\} \setminus \{o_{3,i,j}\}$$

Thank you for your answers. 
It is clear for me now.
 A: What you have described is essentially a prime sieve with a 6th order wheel, so it will indeed contain exactly the prime numbers congruent to $\pm1\mod6$, that is, all primes except for $2$ and $3$. I would not call it a "formula" for prime numbers, but you could certainly use it to implement an algorithm that generates primes up to a given limit (ie. the Sieve of Eratosthenes). 
A: Your equation for the primes is nearly true - it excludes $2$ and $3$, but is otherwise the prime numbers. The only lemma one needs to prove this is the following:

If $n$ is an integer not divisible by $2$ or $3$, then it is either of the form $6k+1$ or $6k-1$.

This tells you that every prime other than those two is in either $\{a_k\}$ or $\{b_k\}$. It also tells you that, for any composite number $n=6k\pm 1$ with $n=ab$, we have that $a$ and $b$ are both of the form $6k\pm 1$ as well, since $2$ and $3$ do not divide $n$ and thus do not divide any divisor of $n$ as well.
This is mostly just a way to speed up the sieve of Eratosthenes, since, to check if a number is prime, you still have to check whether a bunch of numbers divide it. That is, checking whether a given number is in $\{o_{1,i,j}\}$ requires considerable computation. In particular, it's not obvious from this formulation that there's even infinitely many primes, so it seems a bit much to say this is a formula for the primes. The main difficulty in calculating primes is that the structures of the sets $\{o_{n,i,j}\}$ are difficult to describe.
A: Hope this helps with regards to whether it was previously found:
2 and 3 might be referred to as the two "forcibly prime numbers" since there are no integers greater than 1 and less than or equal to their respective square roots. Not a single trial division ever needs to be done for 2 or 3, so they are disqualified from the outset from any attempt to belong to the set of composite numbers. 2 and 3 are thus the only consecutive primes. Since any further prime needs to be coprime to both 2 and 3, they must be congruent to 5 or 1 (mod 2*3) and thus must all be of the form (2*3)*k -/+ 1 with k >= 1. When both (2*3)*k - 1 and (2*3)*k + 1 are prime for a given k >= 1, they are referred to as twin primes (3 and 5 being the only twin primes of the form (2*2)*k - 1 and (2*2)*k + 1). - Daniel Forgues, Mar 19 2010 Link - OEIS.org
