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%I #89 Apr 13 2022 01:16:17
%S 2,5,8,11,14,17,20,23,25,28,31,34,37,40,43,46,49,51,54,57,60,63,66,69,
%T 72,75,77,80,83,86,89,92,95,98,100,103,106,109,112,115,118,121,124,
%U 126,129,132,135,138,141,144,147,150,152,155,158,161,164,167,170,173,176,178,181
%N a(n) = ceiling(2/(2^(1/n)-1)).
%C For n >= 2, a(n) is the 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 Coincides with floor( 2*n/(log 2) ) for all n from 1 to 777451915729367 but differs at 777451915729368. See A129935.
%C The first few values of n for which this sequence differs from floor( 2*n/(log 2) ) were found by _Dean Hickerson_ in 2002. - _N. J. A. Sloane_, Apr 30 2014
%C The sequence floor( log(n)/(2*log(2)) ) is mentioned by Erdős and Selfridge (1973). This sequence begins 0,0,0,1,1,1,1,... = 0 (3 times), 1 (12 times), 2 (48 times), 3 (192 times), 4 (768 times), ..., and grows too slowly to have its own entry. It is related to the game studied by Hales and Jewett (1963). - _N. J. A. Sloane_, Dec 02 2016
%D S. Golomb, "Martin Gardner and Tictacktoe," in Demaine, Demaine, and Rodgers, eds., A Lifetime of Puzzles, A K Peters, 2008, pp. 293-301.
%D 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 Dean Hickerson in 2002, and J. Buhler in 2004, and is reported in S. Golomb (2008).
%D Dean Hickerson, Email to Jon Perry and N. J. A. Sloane, Dec 16 2002. Gives first three terms of A129935: 777451915729368, 140894092055857794, 1526223088619171207, as well as five later terms. - _N. J. A. Sloane_, Apr 30 2014
%H Vincenzo Librandi, <a href="/A078608/b078608.txt">Table of n, a(n) for n = 1..1000</a>
%H Max Alekseyev and others, <a href="http://dxdy.ru/topic6838.html">Integer Parts</a> [in Russian]
%H Max Alekseyev and others, <a href="/A078608/a078608_1.pdf">Integer Parts</a> [in Russian] [Cached copy in pdf format]
%H Art of Problem Solving, <a href="http://www.artofproblemsolving.com/Forum/viewtopic.php?t=140737">Logarithmic Identity</a>
%H Paul Erdős and John L. Selfridge, <a href="https://doi.org/10.1016/0097-3165(73)90005-8">On a combinatorial game</a>, Journal of Combinatorial Theory, Series A 14.3 (1973): 298-301.
%H S. W. Golomb and A. W. Hales, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.116.1297">Hypercube Tic-Tac-Toe</a>
%H A. W. Hales and R. I. Jewett, <a href="https://doi.org/10.1090/S0002-9947-1963-0143712-1">Regularity and Positional Games</a>, Transactions of the American Mathematical Society, vol. 106, no. 2, Feb. 1963, 222-229.
%H K. O'Bryant, <a href="http://arxiv.org/abs/1410.2927">The sequence of fractional parts of roots</a>, arXiv preprint arXiv:1410.2927 [math.NT], 2014-2015.
%H <a href="/index/Se#sequences_which_agree_for_a_long_time">Index entries for sequences which agree for a long time but are different</a>
%t Table[(Ceiling[2/(2^(1/n)-1)]), {n, 1, 100}] (* _Vincenzo Librandi_, May 01 2014 *)
%o (PARI) for (n=2,50, x=2; while (2*x^n<=((x+2)^n),x++); print1(x","))
%o (Haskell)
%o a078608 = ceiling . (2 /) . (subtract 1) . (2 **) . recip . fromIntegral
%o -- _Reinhard Zumkeller_, Mar 27 2015
%Y Cf. A078607, A078609, A129935.
%K nonn
%O 1,1
%A _Jon Perry_, Dec 09 2002
%E Edited by _Dean Hickerson_, Dec 17 2002
%E Revised by _N. J. A. Sloane_, Jun 07 2007
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