Isolated & Non-Isolated Divisors (or whatever they are called)

[Leroy Quet]

It's new to me, but I'm no expert. It is another explicit attempt at
connecting Additive Number Theory with Multiplicative Number Theory.
This is where most of the hardest problems reside, as with Goldbach's
Conjecture, the Twin Primes Conjecture, and Fermat's Last Theorem.

On 9/23/07, Leroy Quet <qq-quet at mindspring.com> wrote:
> Let an "isolated divisor" of n be a positive divisor, k, of n where
> neither (k-1) nor (k+1) divides n.
>
> Let a "non-isolated divisor" of n be a positive divisor, k, of n where
> either (k-1) and/or (k+1) divides n. In other words, a non-isolated
> divisor of n is a positive integer, k, where k(k-1) and/or k(k+1) divides
> n.
>
> For example, the positive divisors of 20 are 1,2,4,5,10,20. Of these, 1
> and 2 are adjacent, and 4 and 5 are adjacent. So the isolated divisors of
> 20 are 10 and 20. While the non-isolated divisors of 20 are 1,2,4,5.
>
> (And of course, every positive divisor of n is either isolated or
> non-isolated and is not both.)
>
> I have submitted several sequences regarding these divisors. But that
> brings me to my question. I have a hard time believing that this concept
> is original. (I have come up with the terms "isolated divisor" and
> "non-isolated divisor".) What I wonder is, are these types of divisors
> already named something else, and I have made a mistake by submitting a
> number of sequences with my terminology?
> I have tried to find alternative terms for these divisors on-line, but so
> far have had no luck.
>
> Thanks,
> Leroy Quet
>