On the K-th extension of the Sieve of eratosthenes

Abstract

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 k we let the initial set Sk consists of natural numbers relatively prime to the first k primes, and find the rules governing the positions of the multiples of the remaining elements.