Generalized perfect numbers

In this note, we continue an approach pursued in an earlier paper of the second author and thereby attempt to produce an improved lower bound for the sum $I(q^k) + I(n^2)$, where $q^k n^2$ is an odd perfect number with special prime $q$ and $I(x)$ is the abundancy index of the positive integer $x$. In particular, this yields an upper bound for $k$.

In this note, we show that if $N$ is an odd perfect number and $q^{\alpha}$ is some prime power exactly dividing it, then $\sigma(N/q^{\alpha})/q^{\alpha}>5$. In general, we also show that if $\sigma(N/q^{\alpha})/q^{\alpha}<K$, where $K$ is any constant, then $N$ is bounded by some function depending on $K$.

Some new results concerning the equation $\sigma(N)=aM, \sigma(M)=bN$ are proved. As a corollary, there are only finitely many odd superperfect numbers with a fixed number of distinct prime factors.

Using an extension of the abundancy index to imaginary quadratic rings with unique factorization, we define what we call $n$-powerfully perfect numbers in these rings. This definition serves to extend the concept of perfect numbers that have been defined and studied in the integers. We investigate the properties of $2$-powerfully perfect numbers in the rings $\mathcal O_{\mathbb{Q}(\sqrt{-1})}$, $\mathcal O_{\mathbb{Q}(\sqrt{-2})}$, and $\mathcal O_{\mathbb{Q}(\sqrt{-7})}$, t...

One of the many number theoretic topics investigated by the ancient Greeks was perfect numbers, which are positive integers equal to the sum of their proper positive integral divisors. Mathematicians from Euclid to Euler investigated these mysterious numbers. We present results on perfect numbers in the ring of Eisenstein integers.

Let $N$ be an odd perfect number. Then, Euler proved that there exist some integers $n, \alpha$ and a prime $q$ such that $N = n^{2}q^{\alpha}$, $q \nmid n$, and $q \equiv \alpha \equiv 1 \bmod 4$. In this note, we prove that the ratio $\frac{\sigma(n^{2})}{q^{\alpha}}$ is neither a square nor a square times a single prime unless $\alpha = 1$. It is a direct consequence of a certain property of the Diophantine equation $2ln^{2} = 1+q+ \cdots +q^{\alpha}$, where $l$ denotes on...

We extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers.

Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We say that $n$ is perfect if $\sigma(n) = 2 n$. Currently there are no known odd perfect numbers. It is known that if an odd perfect number exists, then it must be of the form $N = p^\alpha \prod_{j=1}^k q_j^{2 \beta_j}$, where $p, q_1, ..., q_k$ are distinct primes and $p \equiv \alpha\equiv 1 \pmod{4}$. Define the total number of prime factors of $N$ as $\Omega(N) := \alpha + 2 \sum_{j=1}^k \beta_j$. Sayers sh...

The aim of this note is to show that any even perfect number, other than $6$, can be written as the sum of 5 cubes of natural numbers. We also conjecture that any even perfect number, other than $6$, can be written as the sum of only 3 cubes of natural numbers.

In this note, we present some new results on even almost perfect numbers which are not powers of two. In particular, we show that $2^{r+1} < b$, if ${2^r}{b^2}$ is an even almost perfect number.