On an Extremal Problem concerning Primitive Sequences

A sequence a,< . . . of integers is called primitive if no a divides any other . (a 1 < . . . will always denote a primitive sequence .) It is easy to see that if a i < . . . < a,,, < n then max k = [(n + 1) /2] . The following question seems to be very much more difficult . Put f(n) =niax E 1) , a,, where the maximum is taken over all primitive sequences all of wliose terms are not exceeding n . Determine, or obtain an asymptotic formula for f(n) . Tile explicit determination of f(n) is probably hopeless but we will obtain an asymptotic formula for f(it) . In fact we will prove the following