Talk:Harmonic divisor number

Proposed merger

Harmonic divisor number and Ore's harmonic number are about the same thing. "Harmonic divisor number" seems to be the more common name based on Google. I didn't know these numbers before seeing the articles today. PrimeHunter 01:38, 31 January 2007 (UTC)

My general preference would be to go with what OEIS calls it, over Google or MathWorld which I'd consider less definitive. In this case I think that should be Ore number (currently a redirect to harmonic divisor number) since there is a different and more important usage occupying the harmonic number name. "Ore number" is also the name used in the title of one of the papers cited by OEIS. But I don't feel strongly about the naming issue; any of them will work. As for whether or not to merge, how can there be any question? They are on the same subject, they should be merged. —David Eppstein 02:35, 31 January 2007 (UTC)

I agree the merger is obvious. I just hoped for more knowledgeable people to discuss name and maybe perform the merger. Wikipedia:Naming conventions (common names) suggests to use the most common name and perhaps use Google. Most Google hits on "Ore number" may be unrelated, but there are still many relevant hits and "Ore number" sounds OK to me. "Harmonic divisor number" seems more common (but Google hits may be Wikipedia-influenced) and descriptive. Harmonic number is not about divisors so I don't think the risk of confusion is large. PrimeHunter 13:51, 31 January 2007 (UTC)

I have merged Ore's harmonic number into Harmonic divisor number. PrimeHunter 13:00, 6 March 2007 (UTC)

4 and 12

What's the 4 and 12 doing in the first two formulae? What do they have to do with the harmonic mean? (Besides the fact that they're a factor of the harmonic mean of the divisors, obviously.) —Preceding unsigned comment added by 83.101.8.8 (talk) 07:49, 25 May 2008 (UTC)

The harmonic mean has as its numerator the number of terms whose mean is being taken. There are four terms in the first harmonic mean and twelve terms in the second harmonic mean. —David Eppstein (talk) 08:01, 25 May 2008 (UTC)

Reference for easy theorems

I think that it is better to say "this is easy to show" rather than "look it up here" when the proof is very easy, because this will encourage the reader to do it. A reference might make the reader assume it's nontrivial.Likebox (talk) 19:55, 17 March 2009 (UTC)

A significant refinement of Ore's conjecture

I recently verified that every harmonic divisor number less than 10¹⁴ is a practical number and have conjectured that this is true in general, which is now published on the associated OEIS page, A001599. This is clearly a refinement of Ore's conjecture as every practical number greater than 1 is even, since the only way to represent 2 as a sum of distinct positive integers is with 2 itself. There are other straightforward implications based on Stewart's structure theorem (see the practical numbers page) e.g. if n>1 is a practical number and not a power of 2 (none of which are harmonic divisor numbers), then n=2^{floor(log₂r)}rn, where r is an odd prime and n is a positive integer. Given that the only harmonic divisor numbers of the form q^ap^b - where p and q are primes and a and b are integers not simultaneously equal to 0 - are precisely the even perfect numbers, the conjecture implies that every harmonic divisor number is semiperfect, and primitive semiperfect if and only if it is perfect. Consequently, there should not be any weird harmonic divisor numbers. Put more simply, since every practical number not of the form 2^m is a semiperfect number, we expect that every harmonic divisor number is semiperfect, and so, by definition, not weird.

It may ultimately be easier to prove the part independent of Ore's conjecture - that every even harmonic divisor number is a practical number - which would be somewhat analogous to Euler's proof that every even perfect number is of the form posed by Euclid, but this too has the potential to remain unsolved, at least in full, for a very long time. A particularly easy to observe weak form of the even conjecture follows from Srinivasan's original partial classification of the practical numbers as being divisible by 4 or 6; there are no harmonic divisor numbers congruent to ±2 mod 12.

I would like have the conjecture referenced in appropriate pages (this one and others if there are any), but I don't trust that my writing will be up to Wikipedia standards, and would rather leave the task to a regular contributor. I really appreciate any help you can give me with this. Jaycob Coleman (talk) 11:33, 13 October 2013 (UTC)