Summary: A number \( n>1\) is harmonic if \( \sigma(n)\mid n\tau(n)\), where \( \tau(n)\) and \( \sigma(n)\) are the number of positive divisors of \( n\) and their sum, respectively. It is known that there are no odd harmonic numbers up to \( 10^{15}\). We show here that, for any odd number \( n>10^6, \tau(n)\leq n^{1/3}\). It follows readily that if \( n\) is odd and harmonic, then \( n>p^{3a/2}\) for any prime power divisor \( p^a\) of \( n\), and we have used this in showing that \( n>10^{18}\). We subsequently showed that for any odd number \( n>9 \cdot 10^{17}, \tau(n)\leq n^{1/4}\), from which it follows that if \( n\) is odd and harmonic, then \( n>p^{8a/5}\) with \( p^a\) as before, and we use this improved result in showing that \( n>10^{24}\).