Uncovering Abundant Numbers: My University Research

[RocketSurgery]

Is anyone familiar with Abundant Numbers? As far what kind of work is being done now or any interesting findings.

Me and two professor at my Uni are working on proving (or disproving) some conjectures of mine that have to do with formulas which generate abundant and primitive abundant numbers.

I haven't found too very much on the net about these numbers aside from a few that seem unrelated to our work.

 

[CRGreathouse]

I wrote an undergraduate project on perfect numbers. What are you looking for?

I'm not sure what you mean by "formulas which generate abundant and primitive abundant numbers". Are you trying to find a closed expression for each (primitive) abundant number in order, a formula that produces only/mostly (primitive) abundant numbers, or what? Are the formulas simple expressions with +, -, ^, etc, recurrence relations, implicitly defined functions, or what?

 

[RocketSurgery]

Well we have found simple formulas that generate abundant numbers (no one formula generates all tho. We have shown that they ONLY generate primes tho.
Theres some other work but we are studying these special simple functions first.

 

[RocketSurgery]

The reason I am being so vague is that so far we have a relatively large class of functions all of which seem to be special cases of a formula which does produce the primitive abundants in order. I was just interested if anyone else has studies such formulas or found counter examples. I'm going to run a mathematica program soon to test the one formula.

I'm glad someone else has studied these numbers my professor and I could find hardly any papers on them.

 

[CRGreathouse]

Well we have found simple formulas that generate abundant numbers (no one formula generates all tho. We have shown that they ONLY generate primes tho.
Theres some other work but we are studying these special simple functions first.

I still don't properly understand what you're doing. Making infinitely many abundant numbers from a simple formula is easy -- 6 * 2^n, for example. While perfect numbers have a complex structure that makes them rare, abundant numbers are closed under scalar multiplication (by n > 0).

So you're surely not making a claim that trivial, so where's the meat? Did you find a simple formula that generates a dense set of abundant numbers, perhaps?

The reason I am being so vague is that so far we have a relatively large class of functions all of which seem to be special cases of a formula which does produce the primitive abundants in order. I was just interested if anyone else has studies such formulas or found counter examples. I'm going to run a mathematica program soon to test the one formula.

All abundants in order, and no non-abundant numbers? That would be interesting. Even a high fraction of them would be significant.

I'm glad someone else has studied these numbers my professor and I could find hardly any papers on them.

My undergrad paper on odd perfect numbers had at least 23 citations, and I know I cut some out in the final editing stage. I'd have to imagine there's something out there for generating abundant numbers.

 

[RocketSurgery]

Yes 6*2^n as you have stated does generate a subset of the abundant numbers. What I have found are other formulas which also generate subsets of the abundant numbers.

The important thing is that these subsets aren't subsets of the subset that is created using the formula 6*2^n. So far the union of the 6 or so formulas creates an extremely dense subset of abundant numbers.

Once I work more on this topic over the summer I will give you some concrete details on my research but I want to properly formulate my conjecture before I post anything in case I find a counter example or anything.

I applaud you on your paper on odd perfect numbers. A professor at my school, Dr. Schiffman, whom I have consulted about my research gave a lecture recently on odd abundant numbers (possibly cited your paper when he published some of his results?).

For now though I really need to focus on my finals before I jump back into my research (It's way too damn addicting).

 

[CRGreathouse]

So far the union of the 6 or so formulas creates an extremely dense subset of abundant numbers.

Once I work more on this topic over the summer I will give you some concrete details on my research but I want to properly formulate my conjecture before I post anything in case I find a counter example or anything.

I look forward to hearing about your results, whether negative or positive.