Odd multiperfect numbers of abundancy 4 (Q927709)

A number is multiperfect of abundancy \(k\) (or \(k\)-perfect) if the sum of divisors of \(N\) equals \(kN\). Odd \(k\)-perfect numbers with \(k\geq 2\) supposedly do not exist. For example, if \(N\) is odd and 4-perfect, then \(N\) has at least 22 distinct prime factors. The authors derive, by elementary methods, properties of classes of odd numbers which must be satisfied if they are to be 4-perfect, and conversely, properties of classes which can never be 4-perfect.