

A078608


a(n) = ceiling( 2/(2^(1/n)1)).


5



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
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OFFSET

1,1


COMMENTS

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.
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 Erdös 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

Erdös, Paul, and John L. Selfridge. "On a combinatorial game." Journal of Combinatorial Theory, Series A 14.3 (1973): 298301.
S. Golomb, "Martin Gardner and Tictacktoe," in Demaine, Demaine, and Rodgers, eds., A Lifetime of Puzzles, A K Peters, 2008, pp. 293301.
S. W. Golomb and A. W. Hales, "Hypercube TicTacToe", in "More Games of No Chance", ed. R. J. Nowakowski, MSRI Publications 42, Cambridge University Press, 2002, pp. 167182. 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).
Hales, A. W., and Jewett, R. I., “Regularity and Positional Games,” Transactions of the American Mathematical Society, vol. 106, no. 2, Feb. 1963, 222229.
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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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
S. W. Golomb and A. W. Hales, Hypercube TicTacToe
K. O'Bryant, The sequence of fractional parts of roots, arXiv preprint arXiv:1410.2927, 2014


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

Cf. A078607, A078609, A129935.
Sequence in context: A190336 A276889 A276877 * A189934 A189386 A016789
Adjacent sequences: A078605 A078606 A078607 * A078609 A078610 A078611


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



