Abstract. For any numbers $m$, $n$ with $1< m\le n$ let $\rho_m (n) = \max\sum _{i=1}^r 1/a_i$, where the maximum is taken over all integers $a_1, a_2,\ldots, a_r$ $(r>1)$ which satisfy $1\le a_1 < a_2 < \ldots < a_r\le m$ and $[a_i, a_j]> n$ $(1\le i < j\le r)$. In this paper we derive some properties of $\rho_m (n)$. For example one of our results implies that $\rho_{\pi(n)} (n) \to1$ ($n\to\infty $).

AMS Subject Classification (1991): 11A99

Received March 20, 1996 and in revised form June 26, 1996. (Registered under 6121/2009.)