Why even almost perfect number should not be divisible by 3? A non - almost perfect criterion for even positive integers n≠2k

Abstract

Two of the oldest open problems in elementary number theory are (1) to find a quasi-perfect number and (2) to show that only numbers of the form 2k, k∈Z+ are almost perfect. A positive integer n is quasi perfect if the sum of its positive divisors σ(n) is equal to 2n + 1 where as n is almost perfect if σ(n) is equal to2n − 1. In 1951, Cattaneo showed that quasi perfect numbers cannot be even. Recently, Antalan (2013) showed that almost perfect numbers not of the form 2k must be of the form 2xb2 where x ∈ Z≥0 and b is an odd composite positive integer. Here we give sufficient non - almost perfect criterion for even positive integers ne of the form 2xb2. Particularly we show that ne is automatically not an almost perfect number if it is divisible by 2x and a prime p ≤ 2x+1 − 1. Lastly we state a problem on almost perfect numbers related to what Cattaneo did on quasi perfect number.