A new point of view on perfect and other similar numbers
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Abstract
Omega-perfect numbers (−F -perfect numbers) and [G; ´]-perfect numbers are introduced in the paper. The first of them are the fixed points of an arithmetic function F, while the [G; ´]-perfect numbers are like eigenvectors of an arithmetic function G (if we treat G as an operator) corresponding to ´ (if we treat ´ as an eigenvalue). It is shown that all these numbers are a further generalization of the well-known perfect numbers and of their generalizations and modifications, too. In particular for the so-called g-perfect numbers that were introduced and studied by the authors in 2002 and for the so-called gf -perfect numbers, that are based on Dirichlet's convolution and are introduced for the first time in the present paper, some new properties and results are obtained. Also so-called gf -r-multiperfect numbers are introduced, that generalize harmonic divisor numbers.
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