On the Density of Spoof Odd Perfect Numbers
Grand Duchy of Luxembourg
Rue des Tanneurs 7
L-6790 Grevenmacher
E-mail: uk.laszlo.toth@gmail.com
Received:
Received: 19 February 2021; revised: 19 March 2021; accepted: 23 March 2021; published online: 31 March 2021
DOI: 10.12921/cmst.2021.0000005
Abstract:
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).
Key words:
density, Descartes numbers, lower bound, spoof perfect numbers
References:
[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. Gülog˘lu, 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).
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).
Key words:
density, Descartes numbers, lower bound, spoof perfect numbers
References:
[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. Gülog˘lu, 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).
