Additive properties of even perfect numbers

A perfect number is a number whose divisors add up to twice the number itself. The existence of odd perfect numbers is a millennia-old unsolved problem. This note proposes a proof of the nonexistence of odd perfect numbers. More generally, the same analysis seems to generalize to a proof of the nonexistence of odd multiperfect numbers.

A number is perfect if it is the sum of its proper divisors; here we call a finite group `perfect' if its order is the sum of the orders of its proper normal subgroups. (This conflicts with standard terminology but confusion should not arise.) The notion of perfect group generalizes that of perfect number, since a cyclic group is perfect exactly when its order is perfect. We show that, in fact, the only abelian perfect groups are the cyclic ones, and exhibit some non-abelian ...

A natural number $n$ is called {\it multiperfect} or {\it$k$-perfect} for integer $k\ge2$ if $\sigma(n)=kn$, where $\sigma(n)$ is the sum of the positive divisors of $n$. In this paper, we establish the structure theorem of odd multiperfect numbers analogous as Euler's theorem on odd perfect numbers. We prove the divisibility of the Euler part of odd multiperfect numbers and characterize the forms of odd perfect numbers $n=\pi^\alpha M^2$ such that $\pi\equiv\alpha(\text{mod}...

Let $\sigma(n)$ be the sum of the positive divisors of $n$. A number $n$ is said to be 2-near perfect if $\sigma(n) = 2n +d_1 +d_2 $, where $d_1$ and $d_2$ are distinct positive divisors of $n$. We give a complete description of those $n$ which are 2-near perfect and of the form $n=2^k p^i$ where $p$ is prime and $i \in \{1,2\}$. We also prove related results under the additional restriction where $d_1d_2=n$.

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.

A perfect number is a positive integer n such that n equals the sum of all positive integer divisors of n that are less than n. That is, although n is a divisor of n, n is excluded from this sum. Thus 6 = 1 + 2 + 3 is perfect, but 12 < 1 + 2 + 3 + 4 + 6 is not perfect. An ACL2 theory of perfect numbers is developed and used to prove, in ACL2(r), this bit of mathematical folklore: Even if there are infinitely many perfect numbers the series of the reciprocals of all perfect nu...

In this paper some new ways of generalizing perfect numbers are investigated, numerical results are presented and some conjectures are established.

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...

For a positive integer $n$, if $\sigma(n)$ denotes the sum of the positive divisors of $n$, then $n$ is called a deficient perfect number if $\sigma(n)=2n-d$ for some positive divisor $d$ of $n$. In this paper, we prove some results about odd deficient perfect numbers with four distinct prime factors.

The divisor function $\sigma(n)$ sums the divisors of $n$. We call $n$ abundant when $\sigma(n) - n > n$ and perfect when $\sigma(n) - n = n$. I recently introduced the recursive divisor function $a(n)$, the recursive analog of the divisor function. It measures the extent to which a number is highly divisible into parts, such that the parts are highly divisible into subparts, so on. Just as the divisor function motivates the abundant and perfect numbers, the recursive divisor...