This algorithm was tested in creating correctly the first 10.000 prime number. I'm lazy to do more.

1 Intro

I was looking for patterns in Prime Number and I found and intersting one wich almost could generate prime numbers in a sequential mode. But there was a problem, there was a list of generated number that were not primes, so lets call them anti-primes. Suprisingly this list already existed: A199593. I named them anti-primes :D since this list generates false primes. Conclusion: Generated numbers - anti-primes = Prime Numbers!

So this expository explains an algorithm that generates prime numbers in a for loop.

2 Formulas

2.1 My algorithm

Quite simple, it was fun to test:

| $ x = even => p = 3x-1$

| $ x = odd => p = 3x-2$

2.2 A199593 List

This list/sequence can be defined as all n numbers such that: 3n, 3n-1 and 3n-2 are ALL composite numbers

The formula proposed for this list is: $((1+(-1)^k)((-1)^n)(2n+3)+2k(6n+9+(-1)^n)+((-1)^k)+(12n^2)+36n+29)/4$ and $n,k$ being all natural numbers and 0.

2.3 Final algorithm

Basically the algoritm consists of aplying the simple IF/ELSE from section 2.1 but excluding the indexes that are anti-primes => 2.2 section list

CANT_PRIMES = 10000 // Calculate the first 10k prime numbers
gen = [] // Array to store generated prime numbers

for (i = 0; i < CANT_PRIMES; i++) {
    // If the index is an anti-prime number don't use it
    if (i is anti-prime) {
        continue;
    }

    num = 3 * i - 1; // Asuming it is odd

    // If is even, then we subscarta one more
    if (i % 2 != 0) {
        num--;
    }

    // Add to array
    gen.push(num);
}

2.4 Extra info on A199593

Since I'm not the one that created that list, I didn't dug on how the formula works,Im laizy, thats a reallity I have assumed :), so the use I do in the code is using brute-force to create the A199593 List. || But some of the things I noticed:

• n and k need to be equal to don't loose any anti-prime number. IF this variables are low, some anti-prime numbers won't be generated,thus some generated numbers will not be primes!
• Remove all the 0's from N,K and CANT_PRIMES and then. n*k> CANT_PRIMES must be true so you don't loose any anti-prime numbers.

Extra Use of the BASIC formula

This algorithm can be changed so it checks if a number is prime. Basically get $x-1$ and $x-2$, if one of them is divisible by 3 and the resulting number is not anti-prime then $x$ is a prime number