Additive properties of even perfect numbers

Much recent progress has been made concerning the probable existence of Odd Perfect Numbers, forming part of what has come to be known as Sylvester's Web Of Conditions. This paper proves some results concerning certain properties of the sums of reciprocals of the factors of odd perfect numbers, or, in more technical terms, the properties of the sub-sums of \sigma_{-1} (n). By this result, it also establishes strong bounds on the prime factors of odd perfect numbers using the ...

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.

We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies $\sigma(N)=\frac{n}{d}N$, then N has a prime factor smaller than C, where C is an effective computable constant depending only on s, n, S.

The aim of the present article is to explore the possibilities of representing positive integers as sums of other positive integers and highlight certain fundamental connections between their multiplicative and additive properties. In particular, we shall be concerned with the representation of positive integers as arithmetic series of the simplest kind, i.e., either as sums of successive odd positive numbers, or as sums of successive even positive numbers (both treated as Pr...

Let $\sigma(n)$ to be the sum of the positive divisors of $n$. A number is non-deficient if $\sigma(n) \geq 2n$. We establish new lower bounds for the number of distinct prime factors of an odd non-deficient number in terms of its second smallest, third smallest and fourth smallest prime factors. We also obtain tighter bounds for odd perfect numbers. We also discuss the behavior of $\sigma(n!+1)$, $\sigma(2^n+1)$, and related sequences.

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.

We call positive integer n a near-perfect number, if it is sum of all its proper divisors, except of one of them ("redundant divisor"). We prove an Euclid-like theorem for near-perfect numbers and obtain some other results for them.

We present lower bounds on the sum and product of the distinct prime factors of an odd perfect number, which provide a lower bound on the size of the odd perfect number as a function of the number of its distinct prime factors.

An odd perfect number $N$ is said to be given in Eulerian form if $N = {q^k}{n^2}$ where $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n) = 1$. Similarly, an even perfect number $M$ is said to be given in Euclidean form if $M = (2^p - 1)\cdot{2^{p - 1}}$ where $p$ and $2^p - 1$ are primes. In this article, we show how simple considerations surrounding the differences between the underlying properties of the Eulerian and Euclidean forms of perfect numbers give ri...

We call $n$ a spoof odd perfect number if $n$ is odd and $n=km$ for two integers $k,m>1$ such that $\sigma(k)(m+1)=2n$, where $\sigma$ is the sum-of-divisors function. In this paper, we show how results analogous to those of odd perfect numbers could be established for spoof odd perfect numbers (otherwise known in the literature as Descartes numbers). In particular, we prove that $k$ is not almost perfect.