On near superperfect numbers, the Goormaghtigh conjecture, and Mertens' theorem

Let $\sigma (n)$ be the sum of the divisors of $n$. Kalita and Saikia defined a number $n$ to be near superperfect if $2n+d=\sigma (\sigma (n))$ for some positive divisor $d$ of $n$. We extend some of their results about near superperfect numbers and connect these results to the Goormaghtigh conjecture and to certain products of primes similar to those which appear in Mertens’ theorem. We also define type II near superperfect numbers, which are those $n$ which satisfy $2n+d=\sigma (\sigma (n))$ for some positive divisor $d$ of $\sigma (n)$, and prove analogous results about these numbers.