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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)
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)
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)
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=2floor(log2r)rn, where r is an odd prime and n is a positive integer. Given that the only harmonic divisor numbers of the form qapb - 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 2m 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)
This article is rated B-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
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no archives yet ( create) |
This page has archives. Sections older than 365 days may be automatically archived by Lowercase sigmabot III when more than 5 sections are present. |
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)
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)
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)
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=2floor(log2r)rn, where r is an odd prime and n is a positive integer. Given that the only harmonic divisor numbers of the form qapb - 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 2m 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)