On Wednesday, February 5, 2014 12:40:13 AM UTC, Wizard-Of-Oz wrote:
> I don't see that that will lead to anything terribly useful unless one > can prove that the list is inifinite which means there is no odd perfect > number. But I can't see that there is a way to prove that it is > inifinite or not OTHER THAN be proving there is no odd perfect number. > So I can't see that it would help us arrive at such a prove, rather it > would be a consquent of it being proven.
It's an interesting question on its own. How often do we get numbers that are how close to being perfect? Are there any odd numbers x other than x = 1, 3, 5 and 9 where the difference between the sum of divisors of x and 2x is less than 6? There are a few where the difference is exactly 6. Are there values k where the difference between the sum of divisors of x and 2x is less than k infinitely many times? If yes, which k?