ISSN 0253-2778

CN 34-1054/N

Open AccessOpen Access JUSTC Research Articles:Mathematics

A special class of near-perfect numbers

Funds:  Supported by National Nature Science Foundation of China (11401408) and Project of Science and Technology Department of Sichuan Province (2016JY0134).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2017.11.004
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  • Author Bio:

    LI Jian, male, born in 1992, master. Research field: Number theory. E-mail: lijiansimple@vip.qq.com

  • Corresponding author: LIAO Qunying, PhD/Prof. E-mail: qunyingliao@sicnu.edu.cn
  • Publish Date: 30 November 2017
  • Let α≥2 be an integer, p1 and p2 be odd prime numbers with p12. By using elementary methods and techniques, it was proved that there are no near-perfect numbers of the form 2α-1p12p22 with the redundant divisor d∈{1,p12,p22,p1p2,p1p22,p12p2}, and then an equivalent condition for near-perfect numbers of the form 2α-1p12p22 with the redundant divisor d∈{p1,p2} was obtained. Furthermore, for a fixed positive integer k≥ 2, by generalizing the definition of nearperfect numbers to be k-weakly-near-perfect numbers, it was proved that there are no k-weakly-near-perfect numbers of the form n=2α-1p12p22 when k≥ 3.
    Let α≥2 be an integer, p1 and p2 be odd prime numbers with p12. By using elementary methods and techniques, it was proved that there are no near-perfect numbers of the form 2α-1p12p22 with the redundant divisor d∈{1,p12,p22,p1p2,p1p22,p12p2}, and then an equivalent condition for near-perfect numbers of the form 2α-1p12p22 with the redundant divisor d∈{p1,p2} was obtained. Furthermore, for a fixed positive integer k≥ 2, by generalizing the definition of nearperfect numbers to be k-weakly-near-perfect numbers, it was proved that there are no k-weakly-near-perfect numbers of the form n=2α-1p12p22 when k≥ 3.
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