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It has been shown in previous
investigations of the combinatorial properties of basic sequences that any cohesive
basic sequence ℬ which is contained in ℳ (the set of all pairs of relatively prime
positive integers) must be large in some sense. To be precise, it has been proved that
if ℬ is a cohesive basic sequence and ℬ⊂ℳ, then Cl(p) is infinite for every prime p,
where Cℬ(p) is the set of prime companions of p in primitive pairs in ℬ While this
implies that ℬ must contain a great many primitive pairs, no specific statement
has been made about the density of ℬ It is reasonable to ask, therefore,
whether there are cohesive basic sequences ℬ, contained in ℳ, with density
δ(ℬ) = 0.
It is shown here that such basic sequences do exist, and a method is given for the
construction of a large class of these sequences.
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