1. Introduction. It is the object of this paper to investigate the function γ(m), the number of representations of m in the form
(1) ![](//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20190725032251341-0351:S0008414X00025943:S0008414X00025943_eqns01.gif?pub-status=live)
where
. It is shown that γ(m) is always equal to the number of odd divisors of m, so that for example γ(2k) = 1, this representation being the number 2k itself. From this relationship the average order of γ(m) is deduced ; this result is given in Theorem 2. By a method due to Kac [2], it is shown in §3 that the number of positive integers
for which γ(m) does not exceed a rather complicated function of n and ω, a real parameter, is asymptotically nD(ω), where D(ω) is the probability integral
![](//static.cambridge.org/content/id/urn%3Acambridge.org%3Aid%3Aarticle%3AS0008414X00025943/resource/name/S0008414X00025943_equ01.gif?pub-status=live)