Disproofs of two conjectures concerning nondeficient numbers

Abstract

A positive integer n is said to be nondeficient if

. Letting the positive divisors of an integer

n > 1 $n> 1$ n greater than 1

be written as

1 = d 0 < d 1 < ⋯ < d k < d k + 1 = n $1 = d_0 < d_1 < \cdots < d_k < d_{k+1} = n$ 1 equals d 0 less than d 1 less than midline horizontal ellipsis less than d Subscript k Baseline less than d Subscript k plus 1 Baseline equals n

, and letting

S $\mathcal {S}$ script upper S

denote a set of integers, if there exist values

λ j ∈ S $\lambda _j \in \mathcal {S}$ lamda Subscript j Baseline element of script upper S

such that

1 + ∑ j = 1 k λ j d j = n $1 + \sum _{j=1}^{k} \lambda _j d_j = n$ 1 plus sigma summation Underscript j equals 1 Overscript k Endscripts lamda Subscript j Baseline d Subscript j Baseline equals n

, then n is said to be an

S $\mathcal {S}$ script upper S

-perfect number . Ross, in 2024, introduced the study of

S $\mathcal {S}$ script upper S

-perfect numbers and concluded with two conjectures that each concern both

{ − 1 , 1 } $\{ -1, 1 \}$ StartSet negative 1 comma 1 EndSet

-perfect numbers and nondeficient numbers. We disprove both of these conjectures.