On the Density of Spoof Odd Perfect Numbers

We study the set S of odd positive integers n with the property 2n/σ(n) − 1 = 1/x, for positive integer x, i.e., the set that relates to odd perfect and odd “spoof perfect” numbers. As a consequence, we find that if D = pq denotes a spoof odd perfect number other than Descartes’ example, with pseudo-prime factor p, then q > 1012. Furthermore, we find irregularities in the ending digits of integers n ∈ S and study aspects of its density, leading us to conjecture that the quantity of numbers in S below k is ∼ 10 log(k).

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Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2021).