On near-perfect and deficient-perfect numbers (Q2868723)

A positive integer is called near-perfect if it is the sum of all but one of its proper divisors. \textit{P. Pollack} and \textit{V. Shevelev} [J. Number Theory 132, No. 12, 3037--3046 (2012; Zbl 1272.11013)] constructed three types of near-perfect numbers. \textit{X.-Z. Ren} and \textit{Y.-G. Chen} [Bull. Aust. Math. Soc. 88, No. 3, 520--524 (2013; Zbl 1320.11003)] showed that all near-perfect numbers with two distinct prime factors are of these types. All these numbers are even. \textit{D. Johnson} gave an explicit example of an odd near-perfect number with four distinct prime factors in Sloane's OEIS. The present authors show that there is no odd near-perfect number with three distinct prime factors.NEWLINENEWLINEA positive integer \(n\) is deficient-perfect with deficiency divisor \(d\) if the sum of all positive divisors of \(n\) is equal to \(n-d\), where \(d\) is a proper divisor of \(n\). The authors show that if \(n\) is a deficient-perfect number having at most two distinct prime factors, then \(n=2^{\alpha}\) with deficiency divisor \(d=1\) or \(n=2^{\alpha}(2^{\alpha+1}+2^s-1)\) with deficiency divisor \(d=2^s\), where \(1\leq s\leq\alpha\), and \(2^{\alpha+1}+2^s-1\) is an odd prime.