一类特殊的near-perfect数

A special class of near-perfect numbers

摘要: 设正整数α≥2,p₁,p₂为奇质数且p₁＜p₂.利用初等的方法和技巧,证明了不存在形如2^a-1p₁²p₂²的以∈1,p₁²,p₂²,p₁P₂,p₁p₂²,p₁²p₂为冗余因子的near-perfect数,并给出存在形如2^a-1p₁²p₂²的以d∈p₁,p₂为冗余因子的near-perfect数的一个等价刻画.进而,给定正整数k≥2,通过推广near-perfect数的定义至k弱near-perfect数,证明了当k≥3时,不存在形如2^α-1p₁²p₂²的以d∈p₁²,p₂²为冗余因子的k弱near-perfect数.

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 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-1p₁²p₂²when k≥ 3.