[seqfan] Re: How special is 26?

Hagen von Eitzen math at von-eitzen.de
Sun Feb 14 16:10:52 CET 2010

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"


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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