Origin of Almost Perfect Numbers

Question

Let $N$ be a positive integer. $N$ is called a perfect number if the sum of its positive divisors denoted by $\sigma(N)=2N$. For example $6$ is a perfect number since:

$\sigma(N)=1+2+3+6=12=2(6)$.

An ample of references on the web gives a discussion on the history/origin of perfect numbers. Some are enumerated below.

1. http://www-groups.dcs.st-and.ac.uk/history/HistTopics/Perfect_numbers.html

2. https://prezi.com/qnmg2ssnq-ja/history-of-perfect-numbers/

If $\sigma(N)=2N-1$ we call $N$ an almost perfect number. I tried to search on the web about how almost perfect numbers originated but I have not seen any one.

My question is: How almost perfect numbers originated? In particular who is the first one who considered it as a topic and why they are created?

Thanks a lot for your help.

Answer 1

I did a search and found something surprising: The only known almost-perfect numbers are the powers of $2$. It appears to be unknown if there are others.

The Wolfram link has some references. OEIS 79 has references as well. The main reference is Singh's Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem.

Answer 2

I think the earliest references on almost perfect numbers are due to Cross, and Jerrard & Temperley:

R. P. Jerrard and Nicholas Temperley, Almost perfect numbers, Math. Mag. 46 (1973), 84–87. MR 0376511 (51 #12686). DOI: 10.2307/2689036

James T. Cross, A note on almost perfect numbers, Math. Mag. 47 (1974), 230–231. MR 0354536 (50 #7014). DOI: 10.2307/2689220

You would have to double-check though, if they refer to almost perfect numbers $M$ as numbers satisfying $\sigma(M) = 2M-1$, where $\sigma$ is the classical sum-of-divisors function.