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Spoof odd perfect numbers. (English) Zbl 1370.11005

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.

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
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