Abstract. For any given positive integer $n$ let $\rho(n) = \max\sum _{i=1}^r 1/a_i $, where the maximum is taken over all integers $a_1, a_2, \cdots,a_r $ which satisfy $1\le a_1 < a_2 < \cdots < a_r \leq n$ and $[a_i, a_j] > n$ ($1\leq i< j\leq n$). In this paper we show that $\rho(n) < 1.0170166 $ for large $n$ and give two conjectures either of which implies Erdős' conjecture $\rho(n)\to1$ ($n\to\infty $).

AMS Subject Classification (1991): 11Y60, 11A99

Keyword(s): positive integer, least common multiple, Erdős conjecture

Received March 3, 1995. (Registered under 5711/2009.)