一类特殊的near-perfect数

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2a-1p21p22的以d∈[1,p21,p22,p1P2,p1p22,p21p2]为冗余因子的near-perfect数,并给出存在形如2a-1p21p22的以d∈[p1,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2α-1p21p22的以d∈[p21,p22]为冗余因子的k弱near-perfect数.%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 2a-1p21p22 with the redundant divisor d ∈ [1,p21,p22,p1p2,p1p22,p21p2],and then an equivalent condition for near-perfect numbers of the form 2a-1 p21p22 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=2a 1p21p22 whenk≥ 3.