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

TL;DR

This paper explores near superperfect numbers, extends previous results, and connects them to the Goormaghtigh conjecture and Mertens' theorem, also introducing and analyzing type II near superperfect numbers.

Contribution

It extends existing results on near superperfect numbers, introduces type II near superperfect numbers, and links these concepts to the Goormaghtigh conjecture and Mertens' theorem.

Findings

Extended results on near superperfect numbers.

Defined and analyzed type II near superperfect numbers.

Connected near superperfect numbers to the Goormaghtigh conjecture and Mertens' theorem.

Abstract

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.