Disproofs of two conjectures concerning nondeficient numbers

Abstract

Abstract A positive integer n is said to be nondeficient if σ ( n ) ≥ 2 n $\sigma (n) \geq 2n$ sigma left parenthesis n right parenthesis greater than or equals 2 n . 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