1,5,19,13,211,7,2059,97,1009 - of interest?

[Gottfried Helms]

Hi Jens -

Am 09.08.05 08:23 schrieb jens at voss-ahrensburg.de:

> Hi Gottfried,
> 
> are you aware of Zsigmondy's Theorem? If not, see
> http://mathworld.wolfram.com/ZsigmondyTheorem.html.

Thanks for the link. No - I wasn't aware of that theorem;
and again thanks for the link in your OEIS-entry. It is
a much interesting article of Zsigmondy, and it is
nice to see, how he argues.

The sequence of "primitive factors" seems to be inaccessible
for simple analytical description; the only progress I see
is, that it can be decomposed for a sequence

  f(n,a,b) = (a^n-b^n)/(a-b)

 with the primefactorization p^e,q^f,... into

   f(p^e*q^f,a,b) = Zs(p^e,a,b)*Zs(q^f,a,b)*Zs(n,a,b)
 with
  Zs(n,a,b)= 1 + n*R(n,a,b)

(maybe not exact with the exponents, but given as a scheme)
where R(n,a,b) forms a sequence (along increasing n), which I
cannot describe in a short form.

> 
> It seems your sequence is just Zs(n, 3, 2).
> 
Yes, indeed.

> 
>> So this sequence may be of interest - on the other hand
>> I doubt it is of interest to add sequences for each
>> combination of a and b in f(n,a,b) .. ;-)
> 
> 
> Well, Zs(n, 3, 2) is indeed not in the OEIS, but Zs(n, k, 1)
> (for k € {2, ..., 7}) is. Being an ex-Collatz-junkie myself,
> I strongly vote for the addition of Zs(n, 3, 2).

:-))

Gottfried

---
Gottfried Helms, Univ Kassel