ON A PROBLEM OF ERDOS CONCERNING PRIMITIVE SEQUENCES

A sequence A = {a(i)} of positive integers a1 < a2 < ... is said to be primitive if no term of A divides any other. Let OMEGA(a) denote the number of prime factors of a counted with multiplicity. Let p(a) denote the least prime factor of a and A(p) denote the set of a is-an-element-of A with p(a) = p. The set A(p) is called homogeneous if there is some integer sp such that either A(p) = empty set or OMEGA(a) = s(p) for all a is-an-element-of A(p). Clearly, if A(p) is homogeneous, then A(p) is primitive. The main result of this paper is that if A is a positive integer sequence such that 1 not member A and each A(p) is homogeneous, then SIGMA(a less-than-or-equal-to n, a is-an-element-of A) 1/a log a less-than-or-equal-to SIGMA(p less-than-or-equal-to n,p prime) 1/p log p for n > 1. This would then partially settle a question of Erd6s who asked if this inequality holds for any primitive sequence A .