Odd perfect numbers: lower bound

The first few even perfect numbersA perfect number is a positive integer $n$ such that $n$ is the sum of its proper divisors. For example $6 = 1 + 2 + 3$. The symbol $\sigma(n)$ is usally used for the sum of all the divisors of a positive integer $n$, so that a number is perfect if and only if $\sigma(n) = 2n$. All known perfect numbers are even, and they correspond to Mersenne primes. These are primes of the form $2^k – 1$. For example, if $k=5$ then $2^5 – 1 = 31$, a prime number. The correspondence between Mersenne primes and even perfect numbers is given by:

Theorem. Every even perfect number is of the form $2^{k-1}(2^k – 1)$, and such a number is perfect if and only if $2^k-1$ is a prime number.

What about odd perfect numbers? No one knows if they exist. However, Euler did prove that an odd perfect number has to be of the form
\[ n = p^aq_1^{2b_1}\cdots q_r^{2b_r}\] where $p,q_1,\dots,q_r$ are distinct odd primes, $p\equiv 1\pmod{4}$ and $\alpha\equiv 1\pmod{4}$. In the paper

Hagis, Peter, Jr. Outline of a proof that every odd perfect number has at least eight prime factors. Math. Comp. 35 (1980), no. 151, 1027—1032. MR0572873

the author has shown that $r \geq 7$. This already gives a pretty lower bound for an odd perfect number! Using Euler's restrictions, we see that an odd perfect number has to be greater than 29276722732257. That's on the order of $10^{13}$. Actually, there's a lot more known about the various factors that a perfect number must have. Using all of them, it is shown in

Brent, R. P.; Cohen, G. L.; te Riele, H. J. J. Improved techniques for lower bounds for odd perfect numbers. Math. Comp. 57 (1991), no. 196, 857—868

that an odd perfect number has to be greater than $10^{300}$.

Abundance

The number $A(n) = \sigma(n) – 2n$ is called the abundance of the number $n$. Of course, $A(n) = 0$ if and only if $n$ is perfect. For odd $n$, there are very few numbers for which $|A(n)|$ is small. Of course, the numbers 1,3,5,6,7,9, and 15 are small and so $|A(n)|$ is small for them. But here are the first few numbers after $n=9$ with $|A(n)| \lt 10$:

n 315 1155 8925 32445 442365
$\sigma(n) – 2n$ -6 -6 6 6 6

In fact, these are the twelve odd numbers with $|A(n)| < 10$ under two million. In contrast, there are 81 even numbers satisfying this inequality.

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