On a Generalization of Perfect Numbers

A Problem Proposal

ABSTRACT. This paper presents a notion of perfect numbers relative to arithmetical functions: an arithmetical function f produces a set of f-perfect numbers. Two among the many examples considered are small “perturbations” of the normal definition; late in these two sequences, odd perfect numbers appear! (Could the situation be similar for the usual perfect numbers?) Also, this paper generalizes amicable pairs and sociable chains. Mysteries and open problems abound for those who like a challenge. There is also Mathematica code to start the reader on his own explorations.

Download the paper (File: fperfect.pdf, about 178 Kb). Appeared in The Journal of Recreational Mathematics 31(3).

A web page version is also available, but is not very well formatted (since it was automatically generated by MS Word).

Here is a paper on an especially intriguing sequence of generalized perfect numbers: Picture-Perfect Numbers and Other Digit-Reversal Diversions.

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©2001 J. L. Pe. Document created on 17 December 2001 by J. L. Pe. Last updated on 4 January 2004.