This probably isn't new and you can find this in many places, but I discovered it on my own without looking it up anywhere so I am pretty proud of myself.
Factoring numbers down to their prime factor fingerprint is a very interesting matter. It can take an enormous amount of time with large numbers, hence there is security for the internet. However with this algorithm computing prime factors of numbers up to some target is fairly quick. The algorithm is O(n3).
We begin with the Sieve of Eratosthenes, solving for 1 < x <= n and modify it so that instead of crossing out numbers we append the prime factors to them. This creates a list of all the unique prime factors for every x. Then we go through all of the composites from smallest to largest and divide the number by its unique prime factors. We then take the quotients prime factors and append it to x's list. The Sieve of Eratosthenes runs in O(nloglogn). The second part of the algorithm runs n times and is division which runs n2, hence the Algorithm is n3 + nloglogn. which is O(n3). The real killer of this algorithm is the space complexity rather than the time complexity.
