A059571

Joshua Zucker joshua.zucker at gmail.com
Thu Jun 7 05:39:32 CEST 2007

http://www.research.att.com/~njas/sequences/A059571 has the keyword
"nonn" but if I understand the conjecture correctly, then the fact
that the conjecture has been disproved shows that it's not "nonn" but
rather "sign" ... eventually.  I doubt we'll ever get enough terms
listed (more than 10^16?) to find the first negative term in there.
So how should it be labeled?  I think "sign" even though all the given
terms are positive.

--Joshua Zucker

Dear Seqfans, When I was up at MIT recently Richard Stanley showed me a
remarkable coincidence, which is now encapsulated in the following sequence:

%I A129935
%S A129935 777451915729368
%N A129935 Numbers n such that ceiling( 2/(2^{1/n}-1) ) is not equal to floor( 2n/(log 2) ).
%C A129935 In latex: when is $\left\lceil \frac{2}{2^{1/n}-1}\right\rceil$ different from $\left\lfloor \frac{2n}{\log 2} \right\rfloor$?
%D A129935 S. W. Golomb and A. W. Hales, Hypercube Tic-Tac-Toe, in More Games of No Chance, ed. R. J. Nowakowski, MSRI Publications 42, Cambridge University Press, 2002, pp. 167-182. Here it is stated that the first counterexample is at n=6847196937, an error due to faulty multiprecision arithmetic.  The correct value was found by J. Buhler in 2004 and is reported in S. Golomb, Martin Gardner and Tictacktoe (unpublished).
%O A129935 1,1
%K A129935 nonn,bref,more
%A A129935 Richard Stanley (rstan(AT)math.mit.edu), Apr 30 2007

The companion entry was already in the OEIS, but has now been revised:

%I A078608
%S A078608 2,5,8,11,14,17,20,23,25,28,31,34,37,40,43,46,49,51,54,57,60,63,66,69,72,
%T A078608 75,77,80,83,86,89,92,95,98,100,103,106,109,112,115,118,121,124,126,129,
%U A078608 132,135,138,141,144,147,150,152,155,158,161,164,167,170,173,176,178,181
%N A078608 a(n) = ceiling( 2/(2^(1/n)-1)).
%C A078608 For n >= 2, a(n) = least positive integer x such that 2*x^n>(x+2)^n.  For example, a(2)=5 as 4^2=16, 5^2=25, 6^2=36 and 7^2=49.
%C A078608 Coincides with floor( 2*n/(log 2) ) for all n from 1 to 777451915729367 but differs at 777451915729368. See A129935.
%o A078608 (PARI) for (n=2,50, x=2; while (2*x^n<=((x+2)^n),x++); print1(x","))
%Y A078608 Cf. A078607, A078609, A129935.
%O A078608 1,1
%K A078608 nonn
%A A078608 Jon Perry (perry(AT)globalnet.co.uk), Dec 09 2002
%E A078608 Edited by Dean Hickerson (dean(AT)math.ucdavis.edu), Dec 17 2002
%E A078608 Revised by njas, Jun 07 2007

Maybe some seqfan can extend the first sequence!

Thanks to Andrew Plewe we have had a lot of discussion recently
about possible duplicates. This example shows once again that