This paper presents an alternative recursive piecewise formula that generates all prime numbers only once and in increasing order using only elementary arithmetic operations, square roots and the floor function. The formula does not require previously known prime numbers as input or a precomputed list of primes. Its operation can be interpreted as an algebraic representation of the sieving mechanism underlying the sieve of Eratosthenes through the use of sets of nested equations.The formula is not claimed to improve upon the computational complexity of the sieve of Eratosthenes. Rather, its purpose is theoretical: it provides an explicit and elementary recursive formulation of the sieve mechanism without factorials, trigonometric functions (like the famous 1964 formula of Willans [1], [2] that tautologically defines the prime numbers), or constants whose construction depends on prior knowledge of the primes. The formula was discovered empirically, and no proof of its correctness is provided in this paper.At the end of the article, a comparison is made between prime numbers and lucky numbers, with a similar set of nested equations generating naturally all lucky numbers.