This paper presents a new algorithm for finding all prime numbers between 2 and n. The algorithm runs in time proportional to n, assuming that multiplication of integers not larger than n can be performed in unit time. It is similar in arithmetic complexity to the algorithm presented by Mairson, but is simpler and more elegant. The algorithm can also be extended to find the prime factorization of all integers between 2 and n in time proportional to n. The algorithm works by removing nonprimes from the set {2, ..., n}. Unlike the Sieve of Eratosthenes, it does not attempt to remove a nonprime that was previously removed, which allows for a linear time complexity. The algorithm operates on sets S that are subsets of {2, ..., n} and requires two operations on these sets. The algorithm is designed to efficiently find prime numbers and can be adapted to factorize integers within the same range. The paper also discusses the implications of this algorithm for programming techniques and data structures.This paper presents a new algorithm for finding all prime numbers between 2 and n. The algorithm runs in time proportional to n, assuming that multiplication of integers not larger than n can be performed in unit time. It is similar in arithmetic complexity to the algorithm presented by Mairson, but is simpler and more elegant. The algorithm can also be extended to find the prime factorization of all integers between 2 and n in time proportional to n. The algorithm works by removing nonprimes from the set {2, ..., n}. Unlike the Sieve of Eratosthenes, it does not attempt to remove a nonprime that was previously removed, which allows for a linear time complexity. The algorithm operates on sets S that are subsets of {2, ..., n} and requires two operations on these sets. The algorithm is designed to efficiently find prime numbers and can be adapted to factorize integers within the same range. The paper also discusses the implications of this algorithm for programming techniques and data structures.