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The Sieve of Eratosthenes has been recently extended by excluding the multiplesof 2, 3, and 5 from the initial set, and finding the additive rules that give the positions of themultiples of the remaining primes. We generalize these results. For a given k we let the initialset Sk consists of natural numbers relatively prime to the first k primes, and find the rulesgoverning the positions of the multiples of the remaining elements

On the K-th extension of the Sieve of eratosthenes