Institute of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, China
Funds:
Supported by National Nature Science Foundation of China (11401408) and Project of Science and Technology Department of Sichuan Province (2016JY0134).
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 nearperfect 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.
Abstract
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 nearperfect 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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