The Sieve of Eratosthenes has been recently extended by excluding the multiples of 2, 3, and 5 from the initial set, and finding the additive rules that give the positions of the multiples of the remaining primes. We generalize these results. For a given <mml:math alttext="$k$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> we let the initial set <mml:math alttext="$S_k $" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>S</mml:mi><mml:mi>k</mml:mi></mml:msub></mml:math> consists of natural numbers relatively prime to the first <mml:math alttext="$k$" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>k</mml:mi></mml:math> primes, and find the rules governing the positions of the multiples of the remaining elements.