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

[Alexander Povolotsky]

Two observations/conjectures (using this terminology)
It appears to me that :

1) unless the number "n" itself is 2, the largest positive divisor of n
(which is n itself)
   will be always an "isolated divisor".

2)
a) for any odd n - if "non-isolated divisors" exist, then for every two
"adjacent"
   "non-isolated divisors" - at least one (of the two) is not a prime
factor.
b) for any even n - same as stated in a) except for n=2
(I am assuming that 2 is the prime number)

Alexander Povolotsky

On 9/23/07, Jonathan Post <jvospost3 at gmail.com> wrote:
>
> 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
> >
>
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