Issues in nonlinear hyperperfect numbers

Abstract:

Hyperperfect numbers (HP) are a generalization of perfect numbers and as such share remarkably similar properties. In this note we show, among other things, that if $m = p_1^{{\alpha _1}}p_2^{{\alpha _2}}$ is 2-HP then ${\alpha _2} = 1$, with ${p_1} = 3$, ${p_2} = {3^{{\alpha _1} + 1}} - 2$; this is in agreement with the structure of the perfect case (1-HP) stating that such a number is of the form $m = p_1^{{\alpha _1}}{p_2}$ with ${p_1} = 2$ and ${p_2} = {2^{{\alpha _1} + 1}} - 1$.