On near superperfect numbers, the Goormaghtigh conjecture, and Mertens' theorem
Satvik Beri, Joshua Zelinsky
TL;DR
This work extends Kalita and Saikia's near superperfect framework by analyzing odd $n$ with the condition that each $\sigma(p_i^{a_i})$ is prime, deriving structural constraints such as $k\ge 3$ and specific prime-factor behavior when $k$ is small, and linking these results to the Goormaghtigh conjecture and Mertens-type products through a Kalita-Saikia function. It proves that odd Kalita–Saikia near superperfect numbers force every exponent $a_i\ge 4$, and provides a key bound $\frac{\sigma(\sigma(p_i^{a_i}))}{p_i^{a_i}} < \frac{p_i}{p_i-1}+\frac{1}{p_i^2}$, which constrains possible factor structures. The paper also introduces a density framework showing near superperfect numbers have natural density zero and reports a substantial computational search up to $1.136\times 10^{11}$ identifying only $8$, $21$, $512$, and the Mersenne primes. In addition, Type II near superperfect numbers (where $d|\sigma(n)$) are defined, their early behavior described, and density-zero results extended to Type II and to hybrid $S_{a,b}$ families, framing a broader yet sparse landscape for these phenomena. Overall, the results blend analytic bounds, conjectural connections, and computational evidence to map the rarity and structure of near superperfect numbers and their variants.
Abstract
Let $σ(n)$ be the sum of the divisors of $n$. Kalita and Saikia defined a number $n$ to be near superperfect if $2n+d=σ(σ(n))$ 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 $2n+d=σ(σ(n))$ for some positive divisor $d$ of $σ(n)$, and prove analogous results about these numbers.