On Special Types of Numbers

Abstract

A positive integer n is said to be multiply perfect number if there is a k such that $\sigma\left(n\right)= kn$, where $k\geq1$. In this paper we survey some results of interest on perfect numbers, multiply perfect numbers, k-hyperperfect numbers, superperfect numbers and k-hyper super perfect numbers. To state some results established earlier we have (1). If $n=3^{k-1}\left(3^k-2\right)$ where $3^k-2$ is prime, then $n$ is a 2-hyperperfect number. (2). If $n=3^{p-1}$ where p and $\frac{3^p-1}{2}$ are primes, then n is a super-hyperperfect number.