Bege, Antal; Fogarasi, Kinga Generalized perfect numbers. (English) Zbl 1192.11003 Acta Univ. Sapientiae, Math. 1, No. 1, 73-82 (2009). A positive integer \(n\) is said to be superperfect if \(\sigma^2(n)=\sigma(\sigma(n))=2n\) and \(k\)-hyperperfect if \(\sigma(n)=\frac{k+1}{k}n+\frac{k-1}{k}\). The authors present some results and conjectures on hyperperfect and super-hyperperfect numbers. For a general account of perfect and related numbers, see J. Sándor and B. Crstici [Handbook of number theory. II. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1079.11001)]. Reviewer: Pentti Haukkanen (Tampere) Cited in 2 Documents MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11Y70 Values of arithmetic functions; tables Keywords:perfect number; superperfect number; \(k\)-hyperperfect number Citations:Zbl 1079.11001 PDFBibTeX XMLCite \textit{A. Bege} and \textit{K. Fogarasi}, Acta Univ. Sapientiae, Math. 1, No. 1, 73--82 (2009; Zbl 1192.11003) Full Text: arXiv EuDML Online Encyclopedia of Integer Sequences: Perfect numbers k: k is equal to the sum of the proper divisors of k. 2-hyperperfect numbers: n = 2*(sigma(n) - n - 1) + 1. Numbers k such that 3^k - 2 is prime. Numbers k such that (3^k - 1)/2 is prime. 6-hyperperfect numbers: n = 6*(sigma(n) - n - 1) + 1. 12-hyperperfect numbers: n = 12*(sigma(n) - n - 1) + 1. Numbers m such that m = 2*sigma(m)/3 - 1.