According to Lemma 3.1 of [Lagarias's paper][1] in which he introduces his criterion for RH, one has $$ e^{H_n} \log H_n \geq e^{\gamma} n \log \log n$$ for any $n \geq 3$. This means that any $n$ which violates your inequality would also yield a counterexample to Robin's inequality $$ \sigma (n) < e^\gamma n \log \log n.$$ Since it is known that the truth of RH implies that $5040$ is the largest counterexample, no counterexamples to $\sigma(n) >e^{H_n}\log H_n$ greater than $5040$ should be found. A computer check then confirms that $n=60$ is the largest counterexample. The interest in Lagarias's criterion lies in its requirement being *more* restrictive than Robin's (and, of course, than the inequality in this question).


  [1]: https://arxiv.org/abs/math/0008177