# [seqfan] Re: How special is 26?

Alonso Del Arte alonso.delarte at gmail.com
Mon Feb 15 19:33:53 CET 2010

```Excellent points, Hagen, I'd like to address them one by one.

1.) That's right, it does not. That requirement came from my initial inquiry
into the topic, and I don't think A069586 needs to be modified to include
it. Nor would I submit a new sequence with that requirement since, aside
from a few values, it would be essentially the same as A069586.
2.) By my understanding of the sequence, any value may be repeated. A
Mathematica program that I wrote using prime powers from 4 to 9840769
confirms all 85 values of the sequence currently given, which has only 18
distinct values and many instances of 4, 8, 9, to name just a few besides 0.
3.) I have not yet studied A025475 in any great detail, but I definitely
should.
4.) Any problem with a definition clause is more than a nit. If there's any
fault in the logic of "or 0 if no such q^l," then it needs to be changed.

Al

On Sun, Feb 14, 2010 at 10:10 AM, Hagen von Eitzen <math at von-eitzen.de>wrote:

> Alonso Del Arte wrote:
> > I'd be quicker to offer Mathworld when I can't point to the OEIS.
> >
> > Regardless, on further thinking on this problem I've found A069586, which
> > differs from what I was looking for in that a(4) = 4 rather than 32 (I
> > wanted the bases to be distinct) and that no a(0) is given, rather than
> > giving 16, which suits the criteria by which I initially rejected 4 for
> > a(4).
> >
> > But I'm not saying that A069586 needs to have a(0) = 16, since as 4^2 it
> > does not count for that sequence's definition. However, it would be nice
> to
> > either fill in the zeroes of that sequence or prove them.
> >
> >
> Hm, the sequence A069586 is somewhat inconsistant in several respects:
> 1.) The current wording of the  definition as is does not necessarily
> imply p != q.
> 2.) The word "next" rules out that any value (except 0) be repeated;
> e.g. the sequence currently says a(9) = a(11) = 16, but only 5^2 is the
> next prime power after 2^4, whereas 3^3 is too late.
> 3.) The crossref to A025475 is in support of both these remarks (i.e.
> that p=q is allowed and no intermediate prime power is allowed)
>
> Also, the wording makes a(0) definitely undefined (or rather  a(0)=0)
> because a prime power necessarily differs from the next one.
> As a final act of nitpicking: the definition of A069586 should rather
> end in "... or 0 if no such p^k exists"
>
> Hagen
>
> P.S.: Admittedly, remark 1.) plays a role only near the very beginning
> of the sequence: No prime square between m = p^k and p^(k+1) >= 2m means
> 2*m/(ln(2m)) ~ pi(sqrt(2m)) = pi(sqrt(m)) ~ m/ln(m)
>
>
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```