Kullback-Leibler divergence and primitive non-deficient numbers

Let $H(n)={\prod }_{p\mathrm{\mid }n}\frac{p}{p-1}$ where $p$ ranges over the primes which divide $n$. It is well known that if $n$ is a primitive non-deficient number, then $H(n)>2$. We examine inequalities of the form $H(n)>2+f(n)$ for various functions $f(n)$ where $n$ is assumed to be primitive non-deficient and connect these inequalities to applying the Kullback-Leibler divergence to different probability distributions on the set of divisors of $n$.