On the quotient between the greatest prime factor and the least prime factor

Abstract

Let b(n) be the greatest prime factor of n, a(n) the least prime factor of n and m an arbitrary but fixed positive integer. In this article we prove the asymptotic formulae n k=2 b(k) m a(k) m ∼ C m n m+1 (m + 1) log n where the constant C m (depending of m) is defined in this article. In particular if m = 1 we obtain n k=2 b(k) a(k) ∼ C 1 n 2 2 log n