Subsequence Sum And Perfect Number

Let r0,r1,·…,rk-1 be positive integers and r =(r0,r1,…,rk-1).Denote a finite sequence of integers by A =(a0,a1,…ak-1)r,where a0<a1<…·<ak-1.A subsequence sum of A is the sum of all terms of a nonempty subsequence of A.Denoted by S(A)the set of all subsequence sums of A.In this paper,for given r =(r0,r1,…,rk-1),we give the lower bound for |S(A)| in terms of r and the numbers of positive,negative integers in A.We also determine the structure of the finite sequence A of integers for which |S(A))is minimal.This generalizes the results of Raj Kumar Mistri,Ram Krishna Pandey and Om Prakash in J.Number Theory.Moreover,we give a correction to a result of their paper.Our results have been published in Int.J.Number Theory.For a positive integer n,let σ3(n)=(?)d3.Let n =2α-1pβ-1,where a>1,β>1 are two integers and p is an odd prime.In this paper,we prove that n|σ3(n)if and only if n is an even perfect number except for 28.This extends a result of Tianxin Cai,Deyi Chen and Yong Zhang in Int.J.Number Theory.Our results have been accepted by Colloq.Math..