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We combine the sieves of [1], [2] with the weighted sieve procedure of [3], Chapter 10, to formulate a general theorem about the incidence of almost-primes in integer sequences; and we illustrate the quality of this machinery with many concrete results about almost-primes representable by polynomials with integer or prime arguments.
Chapter 10.1 of [3] describes a weighted sieve procedure which, in combination with upper and lower sieve estimates of dimension κ > 1, leads to a general result of the following kind: given a finite integer sequence A and a set P of primes, then subject only to some rather weak conditions on the pair A, P, one can assert that A contains a large number of almost-primes Pr — numbers having at most r prime factors counted according to multiplicity — with r relatively small, made up of primes from P. Since [3] is currently out of print and higher dimensional sieves superior to those described in Chapter 7 of [3] are now available (see [1], [2]), we state here an improved version of such a general result and describe various applications to integer sequences generated by reducible polynomials that are superior to those presented in Chapter 10.3 of [3].
Let A be a finite integer sequence whose members are not necessarily all positive or distinct.
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