We shall consider a generalization of the sieve process introduced by W. E. Briggs (1) in 1963. Let A(1) be the sequence {ak(1)}, where ak(1) = k + 1, so that A(1) = {2, 3, 4, … }. Suppose inductively that A(1), A(2), … , A(n) has been defined. 4(n+1) will be defined from A(n) = {a1(n), a2(n), a3(n), …} in the following manner: For each integer t ⩾ 0, choose an arbitrary element αt(n) from the set
where an = an(n), and delete the elements αt(n) from A(n) to form A(n+1). The sequence A is defined to be the sequence {an}. It is also the set-theoretic intersection of all the sequences A(n), n — 1, 2, … . Let
be the class of all sequences that can be generated by this sieve process.