摘要
设正整数α≥2,p_1,p_2为奇质数且p_1<p_2.利用初等的方法和技巧,证明了不存在形如2^(α-1) p_1~2p_2~2的以d∈{1,p_1~2,p_2~2,p_1p_2,p_1p_2~2,p_1~2p_2}为冗余因子的near-perfect数,并给出存在形如2^(α-1) p_1~2p_2~2的以d∈{p_1,p_2}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-1) p_1~2p_2~2的以d∈{p_1~2,p_2~2}为冗余因子的k弱near-perfect数.
Letα≥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-1 p1^2 p2^2 with the redundant divisor d ∈ {1,p1^2,p2^2,p1 p2,p1 p2^2,p1^2 p2},and then an equivalent condition for near-perfect numbers of the form 2^α-1 p1^2 p2^2 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 formn=2^α-1 p1^2 p2^2 when k≥ 3.
基金
Supported by National Nature Science Foundation of China(11401408)
Project of Science and Technology Department of Sichuan Province(2016JY0134)
