On the Density of Spoof Odd Perfect Numbers

• Autorzy: Tóth L.

Identyfikatory

DOI

10.12921/cmst.2021.0000005

• Języki publikacji: EN

Abstrakty

EN

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).

Słowa kluczowe

EN

spoof perfect numbers; Descartes numbers; density; lower bound

• Wydawca: Institute of Bioorganic Chemistry Scientific Publishers OWN, Polish Academy of Sciences
• Czasopismo: Computational Methods in Science and Technology
• Rocznik: 2021
• Tom: Vol. 27, No. 1
• Strony: 25--28
• Opis fizyczny: Bibliogr. 4 poz., rys.

Twórcy

autor

Tóth L.

• Grand Duchy of Luxembourg Rue des Tanneurs 7 L-6790 Grevenmacher

Bibliografia

• [1] J. Voight, On the nonexistence of odd perfect numbers, [In:] MASS selecta, Amer. Math. Soc., Providence, RI, 293–300 (2003).
• [2] P.P. Nielsen, Odd perfect numbers have at least nine distinct prime factors, Math. Comp. 76, 2109–2126 (2007).
• [3] W.D. Banks, A.M. Güloglu, C.W. Nevans, F. Saidak, ˘ Descartes numbers, Anatomy of integers 46, 167–173 (2006).
• [4] S.J. Dittmer, Spoof odd perfect numbers, Math. Comp. 83, 2575–2582 (2014).

Uwagi

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