Odd harmonic numbers exceed $10^{24}$

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 $n>10^6$, $\tau(n)\le n^{1/3}$. 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 $n>9\cdot 10^{17}$, $\tau(n)\le n^{1/4}$, 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 $n>10^{24}$.