A078608 a(n) = ceiling(2/(2^(1/n)-1)).

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

Offset: 1

Author

Jon Perry, Dec 09 2002

Keywords

• nonn

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

• 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
• 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.
• Index entries for sequences which agree for a long time but are different

Crossrefs

Cf. A078607, A078609, A129935.

Programs

Haskell

a078608 = ceiling . (2 /) . (subtract 1) . (2 **) . recip . fromIntegral
-- Reinhard Zumkeller, Mar 27 2015

Mathematica

Table[(Ceiling[2/(2^(1/n)-1)]), {n, 1, 100}] (* Vincenzo Librandi, May 01 2014 *)

PARI

for (n=2,50, x=2; while (2*x^n<=((x+2)^n),x++); print1(x","))

Extensions

Edited by Dean Hickerson, Dec 17 2002

Revised by N. J. A. Sloane, Jun 07 2007