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A078608 a(n) = ceiling(2/(2^(1/n)-1)). 6
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, 75, 77, 80, 83, 86, 89, 92, 95, 98, 100, 103, 106, 109, 112, 115, 118, 121, 124, 126, 129, 132, 135, 138, 141, 144, 147, 150, 152, 155, 158, 161, 164, 167, 170, 173, 176, 178, 181 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
Coincides with floor( 2*n/(log 2) ) for all n from 1 to 777451915729367 but differs at 777451915729368. See A129935.
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
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
REFERENCES
S. Golomb, "Martin Gardner and Tictacktoe," in Demaine, Demaine, and Rodgers, eds., A Lifetime of Puzzles, A K Peters, 2008, pp. 293-301.
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).
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
LINKS
Max Alekseyev and others, Integer Parts [in Russian]
Max Alekseyev and others, Integer Parts [in Russian] [Cached copy in pdf format]
Art of Problem Solving, Logarithmic Identity
Paul Erdős and John L. Selfridge, On a combinatorial game, Journal of Combinatorial Theory, Series A 14.3 (1973): 298-301.
S. W. Golomb and A. W. Hales, Hypercube Tic-Tac-Toe
A. W. Hales and R. I. Jewett, Regularity and Positional Games, Transactions of the American Mathematical Society, vol. 106, no. 2, Feb. 1963, 222-229.
K. O'Bryant, The sequence of fractional parts of roots, arXiv preprint arXiv:1410.2927 [math.NT], 2014-2015.
MATHEMATICA
Table[(Ceiling[2/(2^(1/n)-1)]), {n, 1, 100}] (* Vincenzo Librandi, May 01 2014 *)
PROG
(PARI) for (n=2, 50, x=2; while (2*x^n<=((x+2)^n), x++); print1(x", "))
(Haskell)
a078608 = ceiling . (2 /) . (subtract 1) . (2 **) . recip . fromIntegral
-- Reinhard Zumkeller, Mar 27 2015
CROSSREFS
Sequence in context: A276889 A276877 A329961 * A329988 A189934 A189386
KEYWORD
nonn
AUTHOR
Jon Perry, Dec 09 2002
EXTENSIONS
Edited by Dean Hickerson, Dec 17 2002
Revised by N. J. A. Sloane, Jun 07 2007
STATUS
approved

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Last modified February 26 11:46 EST 2024. Contains 370352 sequences. (Running on oeis4.)