On near-perfect numbers (Q5890627)

A positive integer is said to be near-perfect if it is the sum of all of its proper divisors except for one of them. It is not known whether there are infinitely many near-perfect numbers. \textit{P. Pollack} and \textit{V. Shevelev} [J. Number Theory 132, No. 12, 3037--3046 (2012; Zbl 1272.11013)] presented an upper bound for the number of near-perfect numbers and 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{M. Tang} et al. [Colloq. Math. 133, No. 2, 221--226 (2013; Zbl 1332.11004)] proved that there is no odd near-perfect number with three distinct prime factors. It is known that the only odd near-perfect number up to \(1.4\times 10^{19}\) is \(3^4 7^2 11^2 19^2\), see Sloane's OEIS, A181595. The present authors prove that \(3^4 7^2 11^2 19^2\) is the only odd near-perfect number with four distinct prime divisors. In the proof, the authors go through a large number of cases with elementary methods.