关键词: perfect数, near-perfect数, 冗余因子;k弱near-perfect数

Abstract: Let α≥2 be an integer, p₁ and p₂ be odd prime numbers with p₁<p₂. By using elementary methods and techniques, it was proved that there are no near-perfect numbers of the form 2^α-1p₁²p₂² with the redundant divisor d∈{1,p₁²,p₂²,p₁p₂,p₁p₂²,p₁²p₂}, and then an equivalent condition for near-perfect numbers of the form 2^α-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=2^α-1p₁²p₂² when k≥ 3.

Key words: perfect number, near-perfect number, redundant divisor, k-weakly-near-perfect number