Disproofs of two conjectures concerning nondeficient numbers

Abstract

A positive integer $n$ is said to be nondeficient if $\sigma (n)\ge 2n$. Letting the positive divisors of a positive integer $n$ be written as $1=d_0<d_1<\cdots <d_k<d_{k+1}=n$, and letting $\mathcal{S}$ denote a set of integers, if there exist values ${\lambda }_j\in \mathcal{S}$ such that $1+{\sum }_{j=1}^k{\lambda }_j{d}_j=n$, then $n$ is said to be an $\mathcal{S}$-perfect number. Ross, in 2024, introduced the study of $\mathcal{S}$-perfect numbers, and concluded with two conjectures that each concern both $\{-1,1\}$-perfect numbers and nondeficient numbers. We disprove both of these conjectures.

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