Dittmer, Samuel J. Spoof odd perfect numbers. (English) Zbl 1370.11005 Math. Comput. 83, No. 289, 2575-2582 (2014). Summary: In 1638, Descartes showed that \( 3^2 \cdot 7^2 \cdot 11^2 \cdot 13^2 \cdot 22021^1\) would be an odd perfect number if \( 22021\) were prime. We give a formal definition for such “spoof” odd perfect numbers, and construct an algorithm to find all such integers with a given number of distinct quasi-prime factors. We show that Descartes’ example is the only spoof with less than seven such factors. Cited in 3 Documents MSC: 11A25 Arithmetic functions; related numbers; inversion formulas 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11A67 Other number representations 11D72 Diophantine equations in many variables 11N80 Generalized primes and integers Keywords:abundant; deficient; perfect number; spoof PDFBibTeX XMLCite \textit{S. J. Dittmer}, Math. Comput. 83, No. 289, 2575--2582 (2014; Zbl 1370.11005) Full Text: DOI References: [1] Cook, R. J., Bounds for odd perfect numbers. Number theory, Ottawa, ON, 1996, CRM Proc. Lecture Notes 19, 67-71 (1999), Amer. Math. Soc.: Providence, RI:Amer. Math. Soc. · Zbl 0928.11003 [2] [Dickson] Leonard Eugene Dickson, History of the theory of numbers. Vol. I: Divisibility and primality, Dover, New York, 2005. · Zbl 1214.11001 [3] [Euler] Leonhard Euler, Tractatus de Numerorum Doctrina, Commentationes Arithmeticae Collectae 2 (1849), 514. · Zbl 1460.01020 [4] Heath-Brown, D. R., Odd perfect numbers, Math. Proc. Cambridge Philos. Soc., 115, 2, 191-196 (1994) · Zbl 0805.11005 · doi:10.1017/S0305004100072030 [5] Nielsen, Pace P., An upper bound for odd perfect numbers, Integers, 3, A14, 9 pp. (2003) · Zbl 1085.11003 [6] Nielsen, Pace P., Odd perfect numbers have at least nine distinct prime factors, Math. Comp., 76, 260, 2109-2126 (2007) · Zbl 1142.11086 · doi:10.1090/S0025-5718-07-01990-4 [7] [Nielsen3] Pace P. Nielsen, Odd perfect numbers, Diophantine equations and upper bounds, (preprint 2013) available at math.byu.edu/\textasciitildepace/research.html. \endbiblist · Zbl 1325.11009 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.